Theorem 19.42 2229

Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1957 for a version requiring fewer axioms. See exan 1865 for an immediate version. (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 2228	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑))
3		exancom 1864	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4		ancom 461	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-nf 1787

This theorem is referenced by:  eean  2346  bnj596  32726  bnj916  32913  bnj983  32931