Theorem 19.29 752

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

Assertion

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

Proof of Theorem 19.29

Step	Hyp	Ref	Expression
1		19.20 690	. . . . 5 ⊢ (∀x(φ → ¬ ψ) → (∀xφ → ∀x ¬ ψ))
2		alnex 716	. . . . 5 ⊢ (∀x ¬ ψ ↔ ¬ ∃xψ)
3	1, 2	syl6ib 185	. . . 4 ⊢ (∀x(φ → ¬ ψ) → (∀xφ → ¬ ∃xψ))
4	3	con3i 90	. . 3 ⊢ (¬ (∀xφ → ¬ ∃xψ) → ¬ ∀x(φ → ¬ ψ))
5		df-an 198	. . 3 ⊢ ((∀xφ ∧ ∃xψ) ↔ ¬ (∀xφ → ¬ ∃xψ))
6		exnal 721	. . 3 ⊢ (∃x ¬ (φ → ¬ ψ) ↔ ¬ ∀x(φ → ¬ ψ))
7	4, 5, 6	3imtr4 192	. 2 ⊢ ((∀xφ ∧ ∃xψ) → ∃x ¬ (φ → ¬ ψ))
8		df-an 198	. . 3 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
9	8	biex 733	. 2 ⊢ (∃x(φ ∧ ψ) ↔ ∃x ¬ (φ → ¬ ψ))
10	7, 9	sylibr 175	1 ⊢ ((∀xφ ∧ ∃xψ) → ∃x(φ ∧ ψ))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678

This theorem is referenced by:  19.29r 753  exan 784  r19.29 1295

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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