Theorem 19.24 1985

Description: Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)

Assertion

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

Proof of Theorem 19.24

Step	Hyp	Ref	Expression
1		19.2 1976	. . 3 ⊢ (∀𝑥𝜓 → ∃𝑥𝜓)
2	1	imim2i 16	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3		19.35 1877	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
4	2, 3	sylibr 234	1 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by: (None)