Theorem 19.42 2236

Description: Theorem 19.42 of [Margaris] p. 90. See 19.42v 1962 for a version requiring fewer axioms. See exan 1870 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 2235	. 2 ⊢ (∃𝑥(𝜓 ∧ 𝜑) ↔ (∃𝑥𝜓 ∧ 𝜑))
3		exancom 1869	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
4		ancom 464	. 2 ⊢ ((𝜑 ∧ ∃𝑥𝜓) ↔ (∃𝑥𝜓 ∧ 𝜑))
5	2, 3, 4	3bitr4i 306	1 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Syntax hints:   ↔ wb 209   ∧ wa 399  ∃wex 1787  Ⅎwnf 1791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792

This theorem is referenced by:  eean  2350  bnj596  32392  bnj916  32580  bnj983  32598