Theorem rexeqbi1dv 2817

Description: Equality deduction for restricted existential quantifier. (Contributed by NM, 18-Mar-1997.)

Hypothesis

Ref	Expression
raleqd.1	⊢ (A = B → (φ ↔ ψ))

Assertion

Ref	Expression
rexeqbi1dv	⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))

Distinct variable groups:   x,A   x,B

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

Proof of Theorem rexeqbi1dv

Step	Hyp	Ref	Expression
1		rexeq 2809	. 2 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
2		raleqd.1	. . 3 ⊢ (A = B → (φ ↔ ψ))
3	2	rexbidv 2636	. 2 ⊢ (A = B → (∃x ∈ B φ ↔ ∃x ∈ B ψ))
4	1, 3	bitrd 244	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B ψ))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  frd  5923