Theorem 19.41h 1572

Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1573 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
19.41h.1	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
19.41h	⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

Proof of Theorem 19.41h

Step	Hyp	Ref	Expression
1		19.40 1519	. . 3 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ∃xψ))
2		19.41h.1	. . . . 5 ⊢ (ψ → ∀xψ)
3		id 19	. . . . 5 ⊢ (ψ → ψ)
4	2, 3	exlimih 1481	. . . 4 ⊢ (∃xψ → ψ)
5	4	anim2i 324	. . 3 ⊢ ((∃xφ ∧ ∃xψ) → (∃xφ ∧ ψ))
6	1, 5	syl 14	. 2 ⊢ (∃x(φ ∧ ψ) → (∃xφ ∧ ψ))
7		pm3.21 251	. . . 4 ⊢ (ψ → (φ → (φ ∧ ψ)))
8	2, 7	eximdh 1499	. . 3 ⊢ (ψ → (∃xφ → ∃x(φ ∧ ψ)))
9	8	impcom 116	. 2 ⊢ ((∃xφ ∧ ψ) → ∃x(φ ∧ ψ))
10	6, 9	impbii 117	1 ⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))

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:  19.42h  1574  sbh  1656  sbidm  1728  19.41v  1779  2exeu  1989