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Theorem ovmpodv2 5904
 Description: Alternate deduction version of ovmpo 5906, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodv2.1 (𝜑𝐴𝐶)
ovmpodv2.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpodv2.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpodv2.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
Assertion
Ref Expression
ovmpodv2 (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 ovmpodv2
StepHypRef Expression
1 eqidd 2140 . . 3 (𝜑 → (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅))
2 ovmpodv2.1 . . . 4 (𝜑𝐴𝐶)
3 ovmpodv2.2 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
4 ovmpodv2.3 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
5 ovmpodv2.4 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)
65eqeq2d 2151 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
76biimpd 143 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
8 nfmpo1 5838 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
9 nfcv 2281 . . . . . 6 𝑥𝐴
10 nfcv 2281 . . . . . 6 𝑥𝐵
119, 8, 10nfov 5801 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1211nfeq1 2291 . . . 4 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
13 nfmpo2 5839 . . . 4 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
14 nfcv 2281 . . . . . 6 𝑦𝐴
15 nfcv 2281 . . . . . 6 𝑦𝐵
1614, 13, 15nfov 5801 . . . . 5 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1716nfeq1 2291 . . . 4 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpodf 5902 . . 3 (𝜑 → ((𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
191, 18mpd 13 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
20 oveq 5780 . . 3 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
2120eqeq1d 2148 . 2 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
2219, 21syl5ibrcom 156 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  (class class class)co 5774   ∈ cmpo 5776 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779 This theorem is referenced by: (None)
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