ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ovmpt2dv2 Structured version   GIF version

Theorem ovmpt2dv2 5576
Description: Alternate deduction version of ovmpt2 5578, 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 2038 . . 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 2048 . . . . 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 5513 . . . 4 x(x 𝐶, y 𝐷𝑅)
9 nfcv 2175 . . . . . 6 xA
10 nfcv 2175 . . . . . 6 xB
119, 8, 10nfov 5478 . . . . 5 x(A(x 𝐶, y 𝐷𝑅)B)
1211nfeq1 2184 . . . 4 x(A(x 𝐶, y 𝐷𝑅)B) = 𝑆
13 nfmpt22 5514 . . . 4 y(x 𝐶, y 𝐷𝑅)
14 nfcv 2175 . . . . . 6 yA
15 nfcv 2175 . . . . . 6 yB
1614, 13, 15nfov 5478 . . . . 5 y(A(x 𝐶, y 𝐷𝑅)B)
1716nfeq1 2184 . . . 4 y(A(x 𝐶, y 𝐷𝑅)B) = 𝑆
182, 3, 4, 7, 8, 12, 13, 17ovmpt2df 5574 . . 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 5461 . . 3 (𝐹 = (x 𝐶, y 𝐷𝑅) → (A𝐹B) = (A(x 𝐶, y 𝐷𝑅)B))
2120eqeq1d 2045 . 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 1242   wcel 1390  (class class class)co 5455  cmpt2 5457
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator