Theorem raleqOLD 3408

Description: Obsolete version of raleq 3404 as of 5-May-2023. Equality theorem for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) (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		nfcv 2976	. 2 ⊢ Ⅎ𝑥𝐴
2		nfcv 2976	. 2 ⊢ Ⅎ𝑥𝐵
3	1, 2	raleqf 3396	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142

This theorem is referenced by: (None)