Theorem raleqbidvOLD 3425

Description: Obsolete version of raleqbidv 3403 as of 30-Apr-2023. Equality deduction for restricted universal quantifier. (Contributed by NM, 6-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
raleqbidvOLD.1	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbidvOLD.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
raleqbidvOLD	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvOLD

Step	Hyp	Ref	Expression
1		raleqbidvOLD.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	raleqdv 3417	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜓))
3		raleqbidvOLD.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	3	ralbidv 3199	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))
5	2, 4	bitrd 281	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  ∀wral 3140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2816  df-clel 2895  df-ral 3145

This theorem is referenced by: (None)