Theorem 19.42 1575

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 1573	. 2 ⊢ (∃x(ψ ∧ φ) ↔ (∃xψ ∧ φ))
3		exancom 1496	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4		ancom 253	. 2 ⊢ ((φ ∧ ∃xψ) ↔ (∃xψ ∧ φ))
5	2, 3, 4	3bitr4i 201	1 ⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))

Syntax hints:   ∧ wa 97   ↔ wb 98  Ⅎwnf 1346  ∃wex 1378

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-nf 1347

This theorem is referenced by:  eean  1803  r2exf  2336