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Theorem ovmpt2dv2 5760
Description: Alternate deduction version of ovmpt2 5762, 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 2089 . . 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 2099 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
76biimpd 142 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
8 nfmpt21 5697 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
9 nfcv 2228 . . . . . 6 𝑥𝐴
10 nfcv 2228 . . . . . 6 𝑥𝐵
119, 8, 10nfov 5661 . . . . 5 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1211nfeq1 2238 . . . 4 𝑥(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
13 nfmpt22 5698 . . . 4 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
14 nfcv 2228 . . . . . 6 𝑦𝐴
15 nfcv 2228 . . . . . 6 𝑦𝐵
1614, 13, 15nfov 5661 . . . . 5 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵)
1716nfeq1 2238 . . . 4 𝑦(𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 5758 . . 3 (𝜑 → ((𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
191, 18mpd 13 . 2 (𝜑 → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆)
20 oveq 5640 . . 3 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
2120eqeq1d 2096 . 2 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑆))
2219, 21syl5ibrcom 155 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = 𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  (class class class)co 5634  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639
This theorem is referenced by: (None)
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