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Theorem ovig 5597
 Description: The value of an operation class abstraction (weak version). (Contributed by set.mm contributors, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovig.1 ((x = A y = B z = C) → (φψ))
ovig.2 ((x R y S) → ∃*zφ)
ovig.3 F = {x, y, z ((x R y S) φ)}
Assertion
Ref Expression
ovig ((A R B S C D) → (ψ → (AFB) = C))
Distinct variable groups:   x,y,z,A   x,B,y,z   x,C,y,z   x,R,y,z   x,S,y,z   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   D(x,y,z)   F(x,y,z)

Proof of Theorem ovig
StepHypRef Expression
1 3simpa 952 . 2 ((A R B S C D) → (A R B S))
2 eleq1 2413 . . . . . 6 (x = A → (x RA R))
3 eleq1 2413 . . . . . 6 (y = B → (y SB S))
42, 3bi2anan9 843 . . . . 5 ((x = A y = B) → ((x R y S) ↔ (A R B S)))
543adant3 975 . . . 4 ((x = A y = B z = C) → ((x R y S) ↔ (A R B S)))
6 ovig.1 . . . 4 ((x = A y = B z = C) → (φψ))
75, 6anbi12d 691 . . 3 ((x = A y = B z = C) → (((x R y S) φ) ↔ ((A R B S) ψ)))
8 ovig.2 . . . 4 ((x R y S) → ∃*zφ)
9 moanimv 2262 . . . 4 (∃*z((x R y S) φ) ↔ ((x R y S) → ∃*zφ))
108, 9mpbir 200 . . 3 ∃*z((x R y S) φ)
11 ovig.3 . . 3 F = {x, y, z ((x R y S) φ)}
127, 10, 11ovigg 5596 . 2 ((A R B S C D) → (((A R B S) ψ) → (AFB) = C))
131, 12mpand 656 1 ((A R B S C D) → (ψ → (AFB) = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  (class class class)co 5525  {coprab 5527 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-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fv 4795  df-ov 5526  df-oprab 5528 This theorem is referenced by:  ov2ag  5598
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