Theorem cbvralv 3358

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. See cbvralvw 3232 based on fewer axioms , but extra disjoint variables. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker cbvralvw 3232 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 1909	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1909	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvralv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvral 3356	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2366

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clel 2806  df-nfc 2881  df-ral 3059

This theorem is referenced by:  cbvral2v  3362  cbvral3v  3364