Theorem raleqOLD 3336

Description: Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2138, ax-11 2155, and ax-12 2172. (Revised by Steven Nguyen, 30-Apr-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
raleqOLD	⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem raleqOLD

Step	Hyp	Ref	Expression
1		biidd 262	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜑))
2	1	raleqbi1dv 3334	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542  ∀wral 3062

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 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2725  df-ral 3063  df-rex 3072

This theorem is referenced by: (None)