Theorem rexbida 2630

Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by NM, 6-Oct-2003.)

Hypotheses

Ref	Expression
ralbida.1	⊢ Ⅎxφ
ralbida.2	⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))

Assertion

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

Proof of Theorem rexbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 ⊢ Ⅎxφ
2		ralbida.2	. . . 4 ⊢ ((φ ∧ x ∈ A) → (ψ ↔ χ))
3	2	pm5.32da 622	. . 3 ⊢ (φ → ((x ∈ A ∧ ψ) ↔ (x ∈ A ∧ χ)))
4	1, 3	exbid 1773	. 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃x(x ∈ A ∧ χ)))
5		df-rex 2621	. 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ))
6		df-rex 2621	. 2 ⊢ (∃x ∈ A χ ↔ ∃x(x ∈ A ∧ χ))
7	4, 5, 6	3bitr4g 279	1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃x ∈ A χ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   ∈ wcel 1710  ∃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-11 1746

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

This theorem is referenced by:  rexbidva  2632  rexbid  2634  dfiun2g  4000  fun11iun  5306