Theorem 19.29r 1509

Description: Variation of Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
19.29r	⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		19.29 1508	. 2 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ))
2		ancom 253	. 2 ⊢ ((∃xφ ∧ ∀xψ) ↔ (∀xψ ∧ ∃xφ))
3		exancom 1496	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4	1, 2, 3	3imtr4i 190	1 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃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

This theorem is referenced by:  19.29r2  1510  19.29x  1511  exan  1580  ax9o  1585  equvini  1638  eu2  1941  intab  3635  imadiflem  4921  bj-inex  9362