Theorem nfrexgenw-P6 696

Description: Dual Form of nfrgenw-P6 684 (weakened form).

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

Hypotheses

Ref	Expression
nfrexgenw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
nfrexgenw-P6.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexgenw-P6	⊢ (∃𝑥𝜑 → 𝜑)

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

Proof of Theorem nfrexgenw-P6

Step	Hyp	Ref	Expression
1		nfrexgenw-P6.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	subneg-P3.39 323	. . 3 ⊢ (𝑥 = 𝑥₁ → (¬ 𝜑 ↔ ¬ 𝜑₁))
3		nfrexgenw-P6.2	. . . 4 ⊢ Ⅎ𝑥𝜑
4		nfrneg-P6 688	. . . 4 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
5	3, 4	bimpr-P4.RC 534	. . 3 ⊢ Ⅎ𝑥 ¬ 𝜑
6	2, 5	nfrgenw-P6 684	. 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑)
7		lemma-L5.01a 600	. 2 ⊢ ((∃𝑥𝜑 → 𝜑) ↔ (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
8	6, 7	bimpr-P4.RC 534	1 ⊢ (∃𝑥𝜑 → 𝜑)

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:  qremexw-P6  703  trnsvsubw-P6  710