Theorem exbi 1650

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 118	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	alimi 1501	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓))
3		exim 1645	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
5		biimpr 130	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
6	5	alimi 1501	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑))
7		exim 1645	. . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → (∃𝑥𝜓 → ∃𝑥𝜑))
8	6, 7	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜓 → ∃𝑥𝜑))
9	4, 8	impbid 129	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  exbii  1651  exbidh  1660  exintrbi  1679  19.19  1712  rexrnmpt  5771