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Theorem ovmpodv 7560
Description: Alternate deduction version of ovmpo 7563, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodf.1 (𝜑𝐴𝐶)
ovmpodf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpodf.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpodf.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
Assertion
Ref Expression
ovmpodv (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpodv
StepHypRef Expression
1 ovmpodf.1 . 2 (𝜑𝐴𝐶)
2 ovmpodf.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
3 ovmpodf.3 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
4 ovmpodf.4 . 2 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
5 nfcv 2897 . 2 𝑥𝐹
6 nfv 1909 . 2 𝑥𝜓
7 nfcv 2897 . 2 𝑦𝐹
8 nfv 1909 . 2 𝑦𝜓
91, 2, 3, 4, 5, 6, 7, 8ovmpodf 7559 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  (class class class)co 7404  cmpo 7406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409
This theorem is referenced by:  xpcco  18145  curf12  18190  curf2  18192
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