Theorem cbvralv2 2997

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
cbvralv2.1	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
cbvralv2.2	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
cbvralv2	⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)

Distinct variable groups:   𝑦,𝐴   𝜓,𝑦   𝑥,𝐵   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem cbvralv2

Step	Hyp	Ref	Expression
1		nfcv 2229	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2229	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1467	. 2 ⊢ Ⅎ𝑦𝜓
4		nfv 1467	. 2 ⊢ Ⅎ𝑥𝜒
5		cbvralv2.2	. 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
6		cbvralv2.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
7	1, 2, 3, 4, 5, 6	cbvralcsf 2993	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1290  ∀wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-sbc 2844  df-csb 2937

This theorem is referenced by: (None)