Theorem rexeqbidvOLD 3423

Description: Obsolete version of rexeqbidv 3401 as of 30-Apr-2023. Equality deduction for restricted existential 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
rexeqbidvOLD	⊢ (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 rexeqbidvOLD

Step	Hyp	Ref	Expression
1		raleqbidvOLD.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	rexeqdv 3415	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝐵 𝜓))
3		raleqbidvOLD.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	3	rexbidv 3295	. 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 1531  ∃wrex 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1775  df-cleq 2812  df-clel 2891  df-rex 3142

This theorem is referenced by: (None)