Theorem 19.29x 1514

Description: Variation of Theorem 19.29 of [Margaris] p. 90 with mixed quantification. (Contributed by NM, 11-Feb-2005.)

Assertion

Ref	Expression
19.29x	⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Proof of Theorem 19.29x

Step	Hyp	Ref	Expression
1		19.29r 1512	. 2 ⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓))
2		19.29 1511	. . 3 ⊢ ((∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑦(𝜑 ∧ 𝜓))
3	2	eximi 1491	. 2 ⊢ (∃𝑥(∀𝑦𝜑 ∧ ∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
4	1, 3	syl 14	1 ⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 97  ∀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)