Theorem 19.19 2230

Description: Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Hypothesis

Ref	Expression
19.19.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem 19.19

Step	Hyp	Ref	Expression
1		19.19.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.9 2206	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3		exbi 1845	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
4	2, 3	bitr3id 285	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-nf 1782

This theorem is referenced by: (None)