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Theorem frd 5922
 Description: Founded relationship in natural deduction form. (Contributed by SF, 12-Mar-2015.)
Hypotheses
Ref Expression
frd.1 (φR Fr A)
frd.2 (φX V)
frd.3 (φX A)
frd.4 (φX)
Assertion
Ref Expression
frd (φy X z X (zRyz = y))
Distinct variable groups:   y,R,z   y,X,z
Allowed substitution hints:   φ(y,z)   A(y,z)   V(y,z)

Proof of Theorem frd
Dummy variables a r x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frd.3 . 2 (φX A)
2 frd.4 . 2 (φX)
3 frd.2 . . 3 (φX V)
4 frd.1 . . . 4 (φR Fr A)
5 brex 4689 . . . . . 6 (R Fr A → (R V A V))
6 breq 4641 . . . . . . . . . . 11 (r = R → (zryzRy))
76imbi1d 308 . . . . . . . . . 10 (r = R → ((zryz = y) ↔ (zRyz = y)))
87rexralbidv 2658 . . . . . . . . 9 (r = R → (y x z x (zryz = y) ↔ y x z x (zRyz = y)))
98imbi2d 307 . . . . . . . 8 (r = R → (((x a x) → y x z x (zryz = y)) ↔ ((x a x) → y x z x (zRyz = y))))
109albidv 1625 . . . . . . 7 (r = R → (x((x a x) → y x z x (zryz = y)) ↔ x((x a x) → y x z x (zRyz = y))))
11 sseq2 3293 . . . . . . . . . 10 (a = A → (x ax A))
1211anbi1d 685 . . . . . . . . 9 (a = A → ((x a x) ↔ (x A x)))
1312imbi1d 308 . . . . . . . 8 (a = A → (((x a x) → y x z x (zRyz = y)) ↔ ((x A x) → y x z x (zRyz = y))))
1413albidv 1625 . . . . . . 7 (a = A → (x((x a x) → y x z x (zRyz = y)) ↔ x((x A x) → y x z x (zRyz = y))))
15 df-found 5905 . . . . . . 7 Fr = {r, a x((x a x) → y x z x (zryz = y))}
1610, 14, 15brabg 4706 . . . . . 6 ((R V A V) → (R Fr Ax((x A x) → y x z x (zRyz = y))))
175, 16syl 15 . . . . 5 (R Fr A → (R Fr Ax((x A x) → y x z x (zRyz = y))))
1817ibi 232 . . . 4 (R Fr Ax((x A x) → y x z x (zRyz = y)))
194, 18syl 15 . . 3 (φx((x A x) → y x z x (zRyz = y)))
20 sseq1 3292 . . . . . 6 (x = X → (x AX A))
21 neeq1 2524 . . . . . 6 (x = X → (xX))
2220, 21anbi12d 691 . . . . 5 (x = X → ((x A x) ↔ (X A X)))
23 raleq 2807 . . . . . 6 (x = X → (z x (zRyz = y) ↔ z X (zRyz = y)))
2423rexeqbi1dv 2816 . . . . 5 (x = X → (y x z x (zRyz = y) ↔ y X z X (zRyz = y)))
2522, 24imbi12d 311 . . . 4 (x = X → (((x A x) → y x z x (zRyz = y)) ↔ ((X A X) → y X z X (zRyz = y))))
2625spcgv 2939 . . 3 (X V → (x((x A x) → y x z x (zRyz = y)) → ((X A X) → y X z X (zRyz = y))))
273, 19, 26sylc 56 . 2 (φ → ((X A X) → y X z X (zRyz = y)))
281, 2, 27mp2and 660 1 (φy X z X (zRyz = y))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ≠ wne 2516  ∀wral 2614  ∃wrex 2615  Vcvv 2859   ⊆ wss 3257  ∅c0 3550   class class class wbr 4639   Fr cfound 5894 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-found 5905 This theorem is referenced by:  frds  5935
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