Theorem exbid 1638

Description: Formula-building rule for existential quantifier (deduction form). (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 1541	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		exbid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1636	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1482  ∃wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  mobid  2088  rexbida  2500  rexbid2  2510  rexeqf  2698  opabbid  4108  repizf2  4205  oprabbid  5997