Theorem raleqOLD 3329

Description: Obsolete version of raleq 3316 as of 9-Mar-2025. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2129, ax-11 2146, and ax-12 2163. (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 3327	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∀wral 3055

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-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2718  df-ral 3056  df-rex 3065

This theorem is referenced by: (None)