Theorem nfrim-P6 689

Description: ENF Over Implication.

Hypotheses

Ref	Expression
nfrim-P6.1	⊢ Ⅎ𝑥𝜑
nfrim-P6.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfrim-P6	⊢ Ⅎ𝑥(𝜑 → 𝜓)

Proof of Theorem nfrim-P6

Step	Hyp	Ref	Expression
1		qimeqex-P5-L2 611	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2		nfrim-P6.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		dfnfreealt-P6 683	. . . . 5 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
4	2, 3	bimpf-P4.RC 532	. . . 4 ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
5		nfrim-P6.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
6		dfnfreealt-P6 683	. . . . 5 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
7	5, 6	bimpf-P4.RC 532	. . . 4 ⊢ (∃𝑥𝜓 → ∀𝑥𝜓)
8	4, 7	imsubd-P3.28c.RC 272	. . 3 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
9		qimeqallhalf-P5 609	. . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
10	1, 8, 9	dsyl-P3.25.RC 262	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓))
11		dfnfreealt-P6 683	. 2 ⊢ (Ⅎ𝑥(𝜑 → 𝜓) ↔ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
12	10, 11	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥(𝜑 → 𝜓)

Syntax hints:  ∀wff-forall 8   → wff-imp 10  ∃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

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:  nfrand-P6  690  nfrbi-P6  691  splitelof-P6  778  nfrnfr-P6  821