Theorem exbi 1581

Description: Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
exbi	⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ))

Proof of Theorem exbi

Step	Hyp	Ref	Expression
1		bi1 178	. . . 4 ⊢ ((φ ↔ ψ) → (φ → ψ))
2	1	alimi 1559	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(φ → ψ))
3		exim 1575	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
4	2, 3	syl 15	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xφ → ∃xψ))
5		bi2 189	. . . 4 ⊢ ((φ ↔ ψ) → (ψ → φ))
6	5	alimi 1559	. . 3 ⊢ (∀x(φ ↔ ψ) → ∀x(ψ → φ))
7		exim 1575	. . 3 ⊢ (∀x(ψ → φ) → (∃xψ → ∃xφ))
8	6, 7	syl 15	. 2 ⊢ (∀x(φ ↔ ψ) → (∃xψ → ∃xφ))
9	4, 8	impbid 183	1 ⊢ (∀x(φ ↔ ψ) → (∃xφ ↔ ∃xψ))

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:  exbii  1582  exbidh  1591  exintrbi  1613  19.19  1862