Theorem exbidh 1870

Description: Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-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	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		exbidh.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	alexbii 1835	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
4	1, 3	syl 17	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  exbidv  1924  exbid  2216  bj-elgab  35127  ac6s6  36330