Theorem exbid 1616

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 1519	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3		exbid.2	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	exbidh 1614	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1460  ∃wex 1492

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by:  mobid  2061  rexbida  2472  rexbid2  2482  rexeqf  2670  opabbid  4070  repizf2  4164  oprabbid  5930