Theorem exgennfrw-P6 697

Description: Dual Form of gennfrw-P6 685 (weakened form).

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

Hypotheses

Ref	Expression
exgennfrw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
exgennfrw-P6.2	⊢ (∃𝑥𝜑 → 𝜑)

Assertion

Ref	Expression
exgennfrw-P6	⊢ Ⅎ𝑥𝜑

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

Proof of Theorem exgennfrw-P6

Step	Hyp	Ref	Expression
1		exgennfrw-P6.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	subneg-P3.39 323	. . 3 ⊢ (𝑥 = 𝑥₁ → (¬ 𝜑 ↔ ¬ 𝜑₁))
3		exgennfrw-P6.2	. . . 4 ⊢ (∃𝑥𝜑 → 𝜑)
4		lemma-L5.01a 600	. . . 4 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
5	3, 4	bimpf-P4.RC 532	. . 3 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
6	2, 5	gennfrw-P6 685	. 2 ⊢ Ⅎ𝑥 ¬ 𝜑
7		nfrneg-P6 688	. 2 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
8	6, 7	bimpf-P4.RC 532	1 ⊢ Ⅎ𝑥𝜑

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

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  df-nfree-D6.1 682

This theorem is referenced by: (None)