Theorem rexbid 2634

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 27-Jun-1998.)

Hypotheses

Ref	Expression
ralbid.1	⊢ Ⅎxφ
ralbid.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
rexbid	⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Proof of Theorem rexbid

Step	Hyp	Ref	Expression
1		ralbid.1	. 2 ⊢ Ⅎxφ
2		ralbid.2	. . 3 ⊢ (φ → (ψ ↔ χ))
3	2	adantr 451	. 2 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
4	1, 3	rexbida 2630	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  Ⅎwnf 1544   ∈ wcel 1710  ∃wrex 2616

This proof depends on 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-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-rex 2621

This theorem is used by:  rexbidv  2636