Theorem 19.42 1880

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 1879	. 2 ⊢ (∃x(ψ ∧ φ) ↔ (∃xψ ∧ φ))
3		exancom 1586	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4		ancom 437	. 2 ⊢ ((φ ∧ ∃xψ) ↔ (∃xψ ∧ φ))
5	2, 3, 4	3bitr4i 268	1 ⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.42v  1905  eean  1912  r2exf  2650