Theorem exan 784

Description: Place a conjunct in the scope of an existential quantifier.

Hypothesis

Ref	Expression
exan.1	⊢ (∃xφ ∧ ψ)

Assertion

Ref	Expression
exan	⊢ ∃x(φ ∧ ψ)

Proof of Theorem exan

Step	Hyp	Ref	Expression
1		hbe1 709	. . . . 5 ⊢ (∃xφ → ∀x∃xφ)
2	1	19.27 750	. . . 4 ⊢ (∀x(ψ ∧ ∃xφ) ↔ (∀xψ ∧ ∃xφ))
3		exan.1	. . . . 5 ⊢ (∃xφ ∧ ψ)
4		ancom 333	. . . . 5 ⊢ ((∃xφ ∧ ψ) ↔ (ψ ∧ ∃xφ))
5	3, 4	mpbi 164	. . . 4 ⊢ (ψ ∧ ∃xφ)
6	2, 5	mpgbi 685	. . 3 ⊢ (∀xψ ∧ ∃xφ)
7		19.29 752	. . 3 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ))
8	6, 7	ax-mp 6	. 2 ⊢ ∃x(ψ ∧ φ)
9		exancom 736	. 2 ⊢ (∃x(ψ ∧ φ) ↔ ∃x(φ ∧ ψ))
10	8, 9	mpbi 164	1 ⊢ ∃x(φ ∧ ψ)

Syntax hints:   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  bm1.3ii 1481

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

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