Theorem raleqOLD 3340

Description: Obsolete version of raleq 3323 as of 9-Mar-2025. (Contributed by NM, 16-Nov-1995.) Remove usage of ax-10 2141, ax-11 2157, and ax-12 2177. (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 3338	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2729  df-ral 3062  df-rex 3071

This theorem is referenced by: (None)