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Theorem ovmpt2dv2 5553
 Description: Alternate deduction version of ovmpt2 5555, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2dv2.1 (φA 𝐶)
ovmpt2dv2.2 ((φ x = A) → B 𝐷)
ovmpt2dv2.3 ((φ (x = A y = B)) → 𝑅 𝑉)
ovmpt2dv2.4 ((φ (x = A y = B)) → 𝑅 = 𝑆)
Assertion
Ref Expression
ovmpt2dv2 (φ → (𝐹 = (x 𝐶, y 𝐷𝑅) → (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 2019 . . 3 (φ → (x 𝐶, y 𝐷𝑅) = (x 𝐶, y 𝐷𝑅))
2 ovmpt2dv2.1 . . . 4 (φA 𝐶)
3 ovmpt2dv2.2 . . . 4 ((φ x = A) → B 𝐷)
4 ovmpt2dv2.3 . . . 4 ((φ (x = A y = B)) → 𝑅 𝑉)
5 ovmpt2dv2.4 . . . . . 6 ((φ (x = A y = B)) → 𝑅 = 𝑆)
65eqeq2d 2029 . . . . 5 ((φ (x = A y = B)) → ((A(x 𝐶, y 𝐷𝑅)B) = 𝑅 ↔ (A(x 𝐶, y 𝐷𝑅)B) = 𝑆))
76biimpd 132 . . . 4 ((φ (x = A y = B)) → ((A(x 𝐶, y 𝐷𝑅)B) = 𝑅 → (A(x 𝐶, y 𝐷𝑅)B) = 𝑆))
8 nfmpt21 5490 . . . 4 x(x 𝐶, y 𝐷𝑅)
9 nfcv 2156 . . . . . 6 xA
10 nfcv 2156 . . . . . 6 xB
119, 8, 10nfov 5455 . . . . 5 x(A(x 𝐶, y 𝐷𝑅)B)
1211nfeq1 2165 . . . 4 x(A(x 𝐶, y 𝐷𝑅)B) = 𝑆
13 nfmpt22 5491 . . . 4 y(x 𝐶, y 𝐷𝑅)
14 nfcv 2156 . . . . . 6 yA
15 nfcv 2156 . . . . . 6 yB
1614, 13, 15nfov 5455 . . . . 5 y(A(x 𝐶, y 𝐷𝑅)B)
1716nfeq1 2165 . . . 4 y(A(x 𝐶, y 𝐷𝑅)B) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 5551 . . 3 (φ → ((x 𝐶, y 𝐷𝑅) = (x 𝐶, y 𝐷𝑅) → (A(x 𝐶, y 𝐷𝑅)B) = 𝑆))
191, 18mpd 13 . 2 (φ → (A(x 𝐶, y 𝐷𝑅)B) = 𝑆)
20 oveq 5438 . . 3 (𝐹 = (x 𝐶, y 𝐷𝑅) → (A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B))
2120eqeq1d 2026 . 2 (𝐹 = (x 𝐶, y 𝐷𝑅) → ((A𝐹B) = 𝑆 ↔ (A(x 𝐶, y 𝐷𝑅)B) = 𝑆))
2219, 21syl5ibrcom 146 1 (φ → (𝐹 = (x 𝐶, y 𝐷𝑅) → (A𝐹B) = 𝑆))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  (class class class)co 5432   ↦ cmpt2 5434 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437 This theorem is referenced by: (None)
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