Theorem 19.42 775

Description: Theorem 19.42 of [Margaris] p. 90.

Hypothesis

Ref	Expression
19.42.1	⊢ (φ → ∀xφ)

Assertion

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

Proof of Theorem 19.42

Step	Hyp	Ref	Expression
1		19.42.1	. . 3 ⊢ (φ → ∀xφ)
2	1	19.41 774	. 2 ⊢ (∃x(ψ ∧ φ) ↔ (∃xψ ∧ φ))
3		exancom 736	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x(ψ ∧ φ))
4		ancom 333	. 2 ⊢ ((φ ∧ ∃xψ) ↔ (∃xψ ∧ φ))
5	2, 3, 4	3bitr4 158	1 ⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.42v 966  cbvex2 975

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-or 197  df-an 198  df-ex 679

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