Theorem dfexistsint-P7 960

Description: Necessary Condition for (i.e. "If" part of) Existential Quantifier Definition.

Only this direction is deducible with intuitionist logic. †

Assertion

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

Proof of Theorem dfexistsint-P7

Step	Hyp	Ref	Expression
1		dnegi-P3.29.CL 275	. 2 ⊢ (∃𝑥𝜑 → ¬ ¬ ∃𝑥𝜑)
2		allnegex-P7 958	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
3	2	subneg-P3.39.RC 324	. . 3 ⊢ (¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ¬ ∃𝑥𝜑)
4	3	rcp-NDBIER0 241	. 2 ⊢ (¬ ¬ ∃𝑥𝜑 → ¬ ∀𝑥 ¬ 𝜑)
5	1, 4	syl-P3.24.RC 260	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-false-D2.5 158  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682  df-psub-D6.2 716

This theorem is referenced by:  axL6-P7  961