Theorem 19.29r 1621

Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
19.29r	⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		19.29 1620	. 2 ⊢ ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓 ∧ 𝜑))
2		ancom 266	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3		exancom 1608	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4	1, 2, 3	3imtr4i 201	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.29r2  1622  19.29x  1623  exan  1693  ax9o  1698  equvini  1758  eu2  2070  intab  3875  imadiflem  5297  bj-inex  14744