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Theorem cla4ev 159
 Description: Existential introduction.
Hypotheses
Ref Expression
cla4ev.1 A:∗
cla4ev.2 B:α
cla4ev.3 [x:α = B]⊧[A = C]
Assertion
Ref Expression
cla4ev C⊧(λx:α A)
Distinct variable groups:   x,B   x,C   α,x

Proof of Theorem cla4ev
StepHypRef Expression
1 cla4ev.1 . . . . 5 A:∗
2 cla4ev.3 . . . . 5 [x:α = B]⊧[A = C]
31, 2eqtypi 69 . . . 4 C:∗
43id 25 . . 3 CC
5 cla4ev.2 . . . . 5 B:α
61, 5, 2cl 106 . . . 4 ⊤⊧[(λx:α AB) = C]
73, 6a1i 28 . . 3 C⊧[(λx:α AB) = C]
84, 7mpbir 77 . 2 C⊧(λx:α AB)
91wl 59 . . 3 λx:α A:(α → ∗)
109, 5ax4e 158 . 2 (λx:α AB)⊧(λx:α A)
118, 10syl 16 1 C⊧(λx:α A)
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12  ∃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 This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119  df-ex 121 This theorem is referenced by:  axpow  208  axun  209
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