Theorem exan 1681

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
exan.1	⊢ (∃𝑥𝜑 ∧ 𝜓)

Assertion

Ref	Expression
exan	⊢ ∃𝑥(𝜑 ∧ 𝜓)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		hbe1 1483	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑)
2	1	19.28h 1550	. . 3 ⊢ (∀𝑥(∃𝑥𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
3		exan.1	. . 3 ⊢ (∃𝑥𝜑 ∧ 𝜓)
4	2, 3	mpgbi 1440	. 2 ⊢ (∃𝑥𝜑 ∧ ∀𝑥𝜓)
5		19.29r 1609	. 2 ⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
6	4, 5	ax-mp 5	1 ⊢ ∃𝑥(𝜑 ∧ 𝜓)

Syntax hints:   ∧ wa 103  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bm1.3ii  4103  bdbm1.3ii  13773