Theorem exbi 1495

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 111	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	alimi 1344	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓))
3		exim 1490	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5		bi2 121	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
6	5	alimi 1344	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑))
7		exim 1490	. . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
8	6, 7	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
9	4, 8	impbid 120	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  exbii  1496  exbidh  1505  exintrbi  1524  19.19  1556  rexrnmpt  5310