Theorem exbid 1639

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 1542	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		exbid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1637	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1483  ∃wex 1515

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484

This theorem is referenced by:  mobid  2089  rexbida  2501  rexbid2  2511  rexeqf  2699  opabbid  4109  repizf2  4206  oprabbid  5998