Theorem 19.42 1710

Description: Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
19.42.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
19.42	⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.42

Step	Hyp	Ref	Expression
1		19.42.1	. . 3 ⊢ Ⅎ𝑥𝜑
2	1	19.41 1708	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑))
3		exancom 1630	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4		ancom 266	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑))
5	2, 3, 4	3bitr4i 212	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ∧ wa 104   ↔ wb 105  Ⅎwnf 1482  ∃wex 1514

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483

This theorem is referenced by:  eean  1958  r2exf  2523