Theorem 19.29r 753

Description: Variation of Theorem 19.29 of [Margaris] p. 90.

Assertion

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

Proof of Theorem 19.29r

Step	Hyp	Ref	Expression
1		19.29 752	. 2 ⊢ ((∀xψ ∧ ∃xφ) → ∃x(ψ ∧ φ))
2		ancom 333	. 2 ⊢ ((∃xφ ∧ ∀xψ) ↔ (∀xψ ∧ ∃xφ))
3		exancom 736	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4	1, 2, 3	3imtr4 192	1 ⊢ ((∃xφ ∧ ∀xψ) → ∃x(φ ∧ ψ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  eu2 1023  imadif 2714  kmlem6 3585

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-gen 677

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

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