Theorem exbi 1597

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		biimp 117	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	alimi 1448	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓))
3		exim 1592	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5		biimpr 129	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
6	5	alimi 1448	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑))
7		exim 1592	. . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
8	6, 7	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
9	4, 8	impbid 128	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  exbii  1598  exbidh  1607  exintrbi  1626  19.19  1659  rexrnmpt  5639