Theorem allnegex-P7-L2 957

Description: Lemma for allnegex-P7 958. †

Assertion

Ref	Expression
allnegex-P7-L2	⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ 𝜑)

Proof of Theorem allnegex-P7-L2

Step	Hyp	Ref	Expression
1		ndnfrex1-P7.8 833	. . 3 ⊢ Ⅎ𝑥∃𝑥𝜑
2	1	ndnfrneg-P7.2.RC 875	. 2 ⊢ Ⅎ𝑥 ¬ ∃𝑥𝜑
3		exi-P7.CL 952	. . 3 ⊢ (𝜑 → ∃𝑥𝜑)
4	3	trnsp-P3.31c.RC 286	. 2 ⊢ (¬ ∃𝑥𝜑 → ¬ 𝜑)
5	2, 4	alli-P7 947	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ 𝜑)

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-L11 28  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  df-psub-D6.2 716

This theorem is referenced by:  allnegex-P7  958