Theorem exbid 1773

Description: Formula-building rule for existential quantifier (deduction rule). (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
exbid	⊢ (φ → (∃xψ ↔ ∃xχ))

Proof of Theorem exbid

Step	Hyp	Ref	Expression
1		exbid.1	. . 3 ⊢ Ⅎxφ
2	1	nfri 1762	. 2 ⊢ (φ → ∀xφ)
3		exbid.2	. 2 ⊢ (φ → (ψ ↔ χ))
4	2, 3	exbidh 1591	1 ⊢ (φ → (∃xψ ↔ ∃xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541  Ⅎwnf 1544

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-ex 1542  df-nf 1545

This theorem is referenced by:  mobid  2238  rexbida  2630  rexeqf  2805  opabbid  4625  dfid3  4769  oprabbid  5564