Theorem 19.29r 1512

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 1511	. 2 ⊢ ((∀𝑥𝜓 ∧ ∃𝑥𝜑) → ∃𝑥(𝜓 ∧ 𝜑))
2		ancom 253	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) ↔ (∀𝑥𝜓 ∧ ∃𝑥𝜑))
3		exancom 1499	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4	1, 2, 3	3imtr4i 190	1 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(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:  19.29r2  1513  19.29x  1514  exan  1583  ax9o  1588  equvini  1641  eu2  1944  intab  3644  imadiflem  4978  bj-inex  10027