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Theorem ovig 7285
Description: The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
ovig.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
ovig.2 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
ovig.3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
Assertion
Ref Expression
ovig ((𝐴𝑅𝐵𝑆𝐶𝐷) → (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 1140 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝐴𝑅𝐵𝑆))
2 eleq1 2897 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝐴𝑅))
3 eleq1 2897 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝑆𝐵𝑆))
42, 3bi2anan9 635 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
543adant3 1124 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥𝑅𝑦𝑆) ↔ (𝐴𝑅𝐵𝑆)))
6 ovig.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
75, 6anbi12d 630 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝐴𝑅𝐵𝑆) ∧ 𝜓)))
8 ovig.2 . . . 4 ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑)
9 moanimv 2697 . . . 4 (∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑) ↔ ((𝑥𝑅𝑦𝑆) → ∃*𝑧𝜑))
108, 9mpbir 232 . . 3 ∃*𝑧((𝑥𝑅𝑦𝑆) ∧ 𝜑)
11 ovig.3 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝑅𝑦𝑆) ∧ 𝜑)}
127, 10, 11ovigg 7284 . 2 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (((𝐴𝑅𝐵𝑆) ∧ 𝜓) → (𝐴𝐹𝐵) = 𝐶))
131, 12mpand 691 1 ((𝐴𝑅𝐵𝑆𝐶𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  ∃*wmo 2613  (class class class)co 7145  {coprab 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149
This theorem is referenced by:  addsrpr  10485  mulsrpr  10486
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