Theorem 19.25 1605

Description: Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Assertion

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

Proof of Theorem 19.25

Step	Hyp	Ref	Expression
1		19.35-1 1603	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2	1	alimi 1431	. 2 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → ∀𝑦(∀𝑥𝜑 → ∃𝑥𝜓))
3		exim 1578	. 2 ⊢ (∀𝑦(∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))
4	2, 3	syl 14	1 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1329  ∃wex 1468

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)