Theorem exnegallint-P7 1047

Description: "There exists a negative" Implies "Not for all". †

This direction is deducible with intuitionist logic.

Assertion

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

Proof of Theorem exnegallint-P7

Step	Hyp	Ref	Expression
1		ndnfrex1-P7.8 833	. . . 4 ⊢ Ⅎ𝑥∃𝑥 ¬ 𝜑
2		ndnfrall1-P7.7 832	. . . 4 ⊢ Ⅎ𝑥∀𝑥𝜑
3	1, 2	ndnfrand-P7.4.RC 877	. . 3 ⊢ Ⅎ𝑥(∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑)
4		ndnfrv-P7.1 826	. . 3 ⊢ Ⅎ𝑥⊥
5		rcp-NDASM2of2 194	. . . . 5 ⊢ ((∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑) → ∀𝑥𝜑)
6	5	alle-P7 941	. . . 4 ⊢ ((∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑) → 𝜑)
7	6	nimpoe-P4.4b 380	. . 3 ⊢ ((∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑) → (¬ 𝜑 → ⊥))
8		rcp-NDASM1of2 193	. . 3 ⊢ ((∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑) → ∃𝑥 ¬ 𝜑)
9	3, 4, 7, 8	exe-P7 955	. 2 ⊢ ((∃𝑥 ¬ 𝜑 ∧ ∀𝑥𝜑) → ⊥)
10	9	falsenegi-P4.18 432	1 ⊢ (∃𝑥 ¬ 𝜑 → ¬ ∀𝑥𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132  ⊥wff-false 157  ∃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: (None)