Theorem cbvmpt2v 5604

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 3872, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
cbvmpt2v.1	⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
cbvmpt2v.2	⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)

Assertion

Ref	Expression
cbvmpt2v	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵,𝑥,𝑦,𝑧   𝑤,𝐶,𝑧   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpt2v

Step	Hyp	Ref	Expression
1		nfcv 2219	. 2 ⊢ Ⅎ𝑧𝐶
2		nfcv 2219	. 2 ⊢ Ⅎ𝑤𝐶
3		nfcv 2219	. 2 ⊢ Ⅎ𝑥𝐷
4		nfcv 2219	. 2 ⊢ Ⅎ𝑦𝐷
5		cbvmpt2v.1	. . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸)
6		cbvmpt2v.2	. . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷)
7	5, 6	sylan9eq 2133	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷)
8	1, 2, 3, 4, 7	cbvmpt2 5603	1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷)

Syntax hints:   → wi 4   = wceq 1284   ↦ cmpt2 5534

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  frec2uzrdg  9411  frecuzrdgsuc  9417  resqrexlemfp1  9895  resqrex  9912  sqne2sq  10555