Theorem exbidh 1591

Description: Formula-building rule for existential quantifier (deduction rule). (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 1565	. 2 ⊢ (φ → ∀x(ψ ↔ χ))
4		exbi 1581	. 2 ⊢ (∀x(ψ ↔ χ) → (∃xψ ↔ ∃xχ))
5	3, 4	syl 15	1 ⊢ (φ → (∃xψ ↔ ∃xχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-ex 1542

This theorem is referenced by:  exbidv  1626  exbid  1773  drex2  1968