Theorem exbidh 1667

Description: Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
exbidh.1	⊢ (𝜑 → ∀𝑥𝜑)
exbidh.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Proof of Theorem exbidh

Step	Hyp	Ref	Expression
1		exbidh.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		exbidh.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	alrimih 1522	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		exbi 1657	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400  ∃wex 1545

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exbid  1669  drex2  1785  drex1  1851  exbidv  1878  mobidh  2120