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Theorem ovmpodv 7492
Description: Alternate deduction version of ovmpo 7495, 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 2904 . 2 𝑥𝐹
6 nfv 1916 . 2 𝑥𝜓
7 nfcv 2904 . 2 𝑦𝐹
8 nfv 1916 . 2 𝑦𝜓
91, 2, 3, 4, 5, 6, 7, 8ovmpodf 7491 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1540  wcel 2105  (class class class)co 7337  cmpo 7339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342
This theorem is referenced by:  xpcco  17997  curf12  18042  curf2  18044
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