Theorem 19.39 1886

Description: Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Assertion

Ref	Expression
19.39	⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Proof of Theorem 19.39

Step	Hyp	Ref	Expression
1		19.2 1879	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2	1	imim1i 61	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3		19.35 1794	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	sylibr 223	1 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)