Theorem gennex-P6 738

Description: The WFF '¬ ∃𝑥𝜑' is General For '𝑥'.

See gennexw-P6 679 for a version that requires only FOL axioms.

Assertion

Ref	Expression
gennex-P6	⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)

Proof of Theorem gennex-P6

Step	Hyp	Ref	Expression
1		genall-P6 737	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
2		allnegex-P5 597	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	suballinf-P5 594	. 2 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑥𝜑)
4	1, 2, 3	subimd2-P4.RC 545	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19  ax-L10 27  ax-L12 29

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by: (None)