Theorem 19.19 1660

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 1638	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3		exbi 1598	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
4	2, 3	bitr3id 193	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1347  Ⅎwnf 1454  ∃wex 1486

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 1441  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-4 1504  ax-ial 1528

This theorem depends on definitions:  df-bi 116  df-nf 1455

This theorem is referenced by: (None)