Theorem 19.42h 1574

Description: Theorem 19.42 of [Margaris] p. 90. New proofs should use 19.42 1575 instead. (Contributed by NM, 18-Aug-1993.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.42h.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.42h	⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))

Proof of Theorem 19.42h

Step	Hyp	Ref	Expression
1		19.42h.1	. . 3 ⊢ (φ → ∀xφ)
2	1	19.41h 1572	. 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:   → wi 4   ∧ wa 97   ↔ wb 98  ∀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:  19.42v  1783