Theorem cbvralv 3454

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3451 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbvralvw 3451 when possible. (Contributed by NM, 28-Jan-1997.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbvralv.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvralv	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvralv

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvralv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvral 3447	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wral 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clel 2895  df-nfc 2965  df-ral 3145

This theorem is referenced by:  cbvral2v  3466  cbvral3v  3468  frgrwopreglem5ALT  28103