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Theorem ovmpt2dv2 6692
 Description: Alternate deduction version of ovmpt2 6694, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2dv2.1 (𝜑𝐴𝐶)
ovmpt2dv2.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpt2dv2.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpt2dv2.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
Assertion
Ref Expression
ovmpt2dv2 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpt2dv2
StepHypRef Expression
1 eqidd 2611 . . 3 (𝜑 → (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpt2dv2.1 . . . 4 (𝜑𝐴𝐶)
3 ovmpt2dv2.2 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
4 ovmpt2dv2.3 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
5 ovmpt2dv2.4 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
65eqeq2d 2620 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
76biimpd 218 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
8 nfmpt21 6620 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
9 nfcv 2751 . . . . . 6 𝑥𝐴
10 nfcv 2751 . . . . . 6 𝑥𝐵
119, 8, 10nfov 6575 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1211nfeq1 2764 . . . 4 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
13 nfmpt22 6621 . . . 4 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
14 nfcv 2751 . . . . . 6 𝑦𝐴
15 nfcv 2751 . . . . . 6 𝑦𝐵
1614, 13, 15nfov 6575 . . . . 5 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1716nfeq1 2764 . . . 4 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 6690 . . 3 (𝜑 → ((𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
191, 18mpd 15 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
20 oveq 6555 . . 3 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
2120eqeq1d 2612 . 2 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
2219, 21syl5ibrcom 236 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  coaval  16541  xpcco  16646  marrepval  20187  marrepeval  20188  marepveval  20193  submaval  20206  submaeval  20207  minmar1val  20273  minmar1eval  20274  nbgraop  25952  isuvtx  26016  nbgrval  40560
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