Theorem 19.19 2231

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 2205	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3		exbi 1847	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
4	2, 3	syl5bbr 287	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)