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Theorem axrep 207
Description: Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory. Axiom 5 of [TakeutiZaring] p. 19.
Hypotheses
Ref Expression
axrep.1 |- A:*
axrep.2 |- B:(al -> *)
Assertion
Ref Expression
axrep |- T. |= [(A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) ==> (E.\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]))]
Distinct variable groups:   y,B   al,y   x,y,be   y,z,be

Proof of Theorem axrep
Dummy variable f is distinct from all other variables.
StepHypRef Expression
1 wal 124 . . . . . . 7 |- A.:((al -> *) -> *)
2 wex 129 . . . . . . . . 9 |- E.:((be -> *) -> *)
3 wal 124 . . . . . . . . . . 11 |- A.:((be -> *) -> *)
4 wim 127 . . . . . . . . . . . . 13 |- ==> :(* -> (* -> *))
5 axrep.1 . . . . . . . . . . . . . . 15 |- A:*
65wl 59 . . . . . . . . . . . . . 14 |- \y:be A:(be -> *)
73, 6wc 45 . . . . . . . . . . . . 13 |- (A.\y:be A):*
8 wv 58 . . . . . . . . . . . . . 14 |- z:be:be
9 wv 58 . . . . . . . . . . . . . 14 |- y:be:be
108, 9weqi 68 . . . . . . . . . . . . 13 |- [z:be = y:be]:*
114, 7, 10wov 64 . . . . . . . . . . . 12 |- [(A.\y:be A) ==> [z:be = y:be]]:*
1211wl 59 . . . . . . . . . . 11 |- \z:be [(A.\y:be A) ==> [z:be = y:be]]:(be -> *)
133, 12wc 45 . . . . . . . . . 10 |- (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]):*
1413wl 59 . . . . . . . . 9 |- \y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]):(be -> *)
152, 14wc 45 . . . . . . . 8 |- (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]])):*
1615wl 59 . . . . . . 7 |- \x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]])):(al -> *)
171, 16wc 45 . . . . . 6 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))):*
18 wtru 40 . . . . . . . 8 |- T.:*
19 wex 129 . . . . . . . . 9 |- E.:((al -> *) -> *)
20 wan 126 . . . . . . . . . . 11 |- /\ :(* -> (* -> *))
21 axrep.2 . . . . . . . . . . . 12 |- B:(al -> *)
22 wv 58 . . . . . . . . . . . 12 |- x:al:al
2321, 22wc 45 . . . . . . . . . . 11 |- (Bx:al):*
2420, 23, 7wov 64 . . . . . . . . . 10 |- [(Bx:al) /\ (A.\y:be A)]:*
2524wl 59 . . . . . . . . 9 |- \x:al [(Bx:al) /\ (A.\y:be A)]:(al -> *)
2619, 25wc 45 . . . . . . . 8 |- (E.\x:al [(Bx:al) /\ (A.\y:be A)]):*
2718, 26eqid 73 . . . . . . 7 |- T. |= [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
2827alrimiv 141 . . . . . 6 |- T. |= (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])
2917, 28a1i 28 . . . . 5 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) |= (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])
3029ax-cb1 29 . . . . . 6 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))):*
31 wv 58 . . . . . . . . . . 11 |- y:(be -> *):(be -> *)
3231, 8wc 45 . . . . . . . . . 10 |- (y:(be -> *)z:be):*
3332, 26weqi 68 . . . . . . . . 9 |- [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]:*
3433wl 59 . . . . . . . 8 |- \z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]:(be -> *)
353, 34wc 45 . . . . . . 7 |- (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]):*
3626wl 59 . . . . . . 7 |- \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)]):(be -> *)
37 weq 38 . . . . . . . . . 10 |- = :(* -> (* -> *))
3831, 36weqi 68 . . . . . . . . . . . . 13 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]:*
3938id 25 . . . . . . . . . . . 12 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
4031, 8, 39ceq1 79 . . . . . . . . . . 11 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [(y:(be -> *)z:be) = (\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])z:be)]
4126beta 82 . . . . . . . . . . . 12 |- T. |= [(\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
4238, 41a1i 28 . . . . . . . . . . 11 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [(\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
4332, 40, 42eqtri 85 . . . . . . . . . 10 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
4437, 32, 26, 43oveq1 89 . . . . . . . . 9 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [[(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])] = [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]]
45 weq 38 . . . . . . . . . 10 |- = :((be -> *) -> ((be -> *) -> *))
46 wv 58 . . . . . . . . . 10 |- f:be:be
4737, 46ax-17 95 . . . . . . . . . 10 |- T. |= [(\z:be = f:be) = = ]
4831, 46ax-17 95 . . . . . . . . . 10 |- T. |= [(\z:be y:(be -> *)f:be) = y:(be -> *)]
4926, 46ax-hbl1 93 . . . . . . . . . 10 |- T. |= [(\z:be \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])f:be) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
5045, 31, 46, 36, 47, 48, 49hbov 101 . . . . . . . . 9 |- T. |= [(\z:be [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]f:be) = [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]]
5133, 44, 50leqf 169 . . . . . . . 8 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])] = \z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]]
523, 34, 51ceq2 80 . . . . . . 7 |- [y:(be -> *) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])] |= [(A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]) = (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])]
5326, 26weqi 68 . . . . . . . . 9 |- [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]:*
5453wl 59 . . . . . . . 8 |- \z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]:(be -> *)
55 wv 58 . . . . . . . 8 |- f:(be -> *):(be -> *)
563, 55ax-17 95 . . . . . . . 8 |- T. |= [(\y:(be -> *) A.f:(be -> *)) = A.]
57 weq 38 . . . . . . . . . . 11 |- = :(ga -> (ga -> *))
5857, 55ax-17 95 . . . . . . . . . 10 |- T. |= [(\y:(be -> *) = f:(be -> *)) = = ]
592, 55ax-17 95 . . . . . . . . . . 11 |- T. |= [(\y:(be -> *) E.f:(be -> *)) = E.]
6020, 55ax-17 95 . . . . . . . . . . . . 13 |- T. |= [(\y:(be -> *) /\ f:(be -> *)) = /\ ]
6123, 55ax-17 95 . . . . . . . . . . . . 13 |- T. |= [(\y:(be -> *) (Bx:al)f:(be -> *)) = (Bx:al)]
625, 55, 18hbl1 94 . . . . . . . . . . . . . 14 |- T. |= [(\y:(be -> *) \y:be Af:(be -> *)) = \y:be A]
633, 6, 55, 56, 62hbc 100 . . . . . . . . . . . . 13 |- T. |= [(\y:(be -> *) (A.\y:be A)f:(be -> *)) = (A.\y:be A)]
6420, 23, 55, 7, 60, 61, 63hbov 101 . . . . . . . . . . . 12 |- T. |= [(\y:(be -> *) [(Bx:al) /\ (A.\y:be A)]f:(be -> *)) = [(Bx:al) /\ (A.\y:be A)]]
6524, 55, 64hbl 102 . . . . . . . . . . 11 |- T. |= [(\y:(be -> *) \x:al [(Bx:al) /\ (A.\y:be A)]f:(be -> *)) = \x:al [(Bx:al) /\ (A.\y:be A)]]
6619, 25, 55, 59, 65hbc 100 . . . . . . . . . 10 |- T. |= [(\y:(be -> *) (E.\x:al [(Bx:al) /\ (A.\y:be A)])f:(be -> *)) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
6737, 26, 55, 26, 58, 66, 66hbov 101 . . . . . . . . 9 |- T. |= [(\y:(be -> *) [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]f:(be -> *)) = [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]]
6853, 55, 67hbl 102 . . . . . . . 8 |- T. |= [(\y:(be -> *) \z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]f:(be -> *)) = \z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]]
693, 54, 55, 56, 68hbc 100 . . . . . . 7 |- T. |= [(\y:(be -> *) (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])f:(be -> *)) = (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])]
7026, 55, 66hbl 102 . . . . . . 7 |- T. |= [(\y:(be -> *) \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])f:(be -> *)) = \z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])]
7135, 36, 52, 69, 70clf 105 . . . . . 6 |- T. |= [(\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])) = (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])]
7230, 71a1i 28 . . . . 5 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) |= [(\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])) = (A.\z:be [(E.\x:al [(Bx:al) /\ (A.\y:be A)]) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])]
7329, 72mpbir 77 . . . 4 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) |= (\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)]))
7435wl 59 . . . . 5 |- \y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]):((be -> *) -> *)
7574, 36ax4e 158 . . . 4 |- (\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])])\z:be (E.\x:al [(Bx:al) /\ (A.\y:be A)])) |= (E.\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]))
7673, 75syl 16 . . 3 |- (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) |= (E.\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]))
7776, 18adantl 51 . 2 |- (T., (A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]])))) |= (E.\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]))
7877ex 148 1 |- T. |= [(A.\x:al (E.\y:be (A.\z:be [(A.\y:be A) ==> [z:be = y:be]]))) ==> (E.\y:(be -> *) (A.\z:be [(y:(be -> *)z:be) = (E.\x:al [(Bx:al) /\ (A.\y:be A)])]))]
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   /\ tan 109   ==> tim 111  A.tal 112  E.tex 113
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-ex 121
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