Theorem gennexw-P6 679

Description: The WFF '¬ ∃𝑥𝜑' is General for '𝑥' (weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)'.

Hypothesis

Ref	Expression
gennexw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))

Assertion

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

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝑥,𝑥₁

Proof of Theorem gennexw-P6

Step	Hyp	Ref	Expression
1		gennexw-P6.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	subneg-P3.39 323	. . 3 ⊢ (𝑥 = 𝑥₁ → (¬ 𝜑 ↔ ¬ 𝜑₁))
3	2	genallw-P6 676	. 2 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥∀𝑥 ¬ 𝜑)
4		allnegex-P5 597	. 2 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
5	4	suballinf-P5 594	. 2 ⊢ (∀𝑥∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ ∃𝑥𝜑)
6	3, 4, 5	subimd2-P4.RC 545	1 ⊢ (¬ ∃𝑥𝜑 → ∀𝑥 ¬ ∃𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104  ∃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

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

This theorem is referenced by: (None)