Theorem 19.37 2100

Description: Theorem 19.37 of [Margaris] p. 90. See 19.37v 1910 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
19.37.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem 19.37

Step	Hyp	Ref	Expression
1		19.35 1805	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.37.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	19.3 2069	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
4	3	imbi1i 339	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
5	1, 4	bitri 264	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj900  30999