Theorem 19.25 1517

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 1515	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2	1	alimi 1344	. 2 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → ∀𝑦(∀𝑥𝜑 → ∃𝑥𝜓))
3		exim 1490	. 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 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)