Theorem cbvralv2 3881

Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.)

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 2907	. 2 ⊢ Ⅎ𝑦𝐴
2		nfcv 2907	. 2 ⊢ Ⅎ𝑥𝐵
3		nfv 1917	. 2 ⊢ Ⅎ𝑦𝜓
4		nfv 1917	. 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 3877	1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-sbc 3717  df-csb 3833

This theorem is referenced by: (None)