Theorem nfrimd-P6 815

Description: ENF Over Implication (deductive form).

Hypotheses

Ref	Expression
nfrimd-P6.1	⊢ (𝛾 → Ⅎ𝑥𝜑)
nfrimd-P6.2	⊢ (𝛾 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrimd-P6	⊢ (𝛾 → Ⅎ𝑥(𝜑 → 𝜓))

Proof of Theorem nfrimd-P6

Step	Hyp	Ref	Expression
1		qimeqex-P5 612	. . . . 5 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2	1	rcp-NDBIEF0 240	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓)))
4		nfrimd-P6.1	. . . . 5 ⊢ (𝛾 → Ⅎ𝑥𝜑)
5		dfnfreealt-P6 683	. . . . 5 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
6	4, 5	subimr2-P4.RC 543	. . . 4 ⊢ (𝛾 → (∃𝑥𝜑 → ∀𝑥𝜑))
7		nfrimd-P6.2	. . . . 5 ⊢ (𝛾 → Ⅎ𝑥𝜓)
8		dfnfreealt-P6 683	. . . . 5 ⊢ (Ⅎ𝑥𝜓 ↔ (∃𝑥𝜓 → ∀𝑥𝜓))
9	7, 8	subimr2-P4.RC 543	. . . 4 ⊢ (𝛾 → (∃𝑥𝜓 → ∀𝑥𝜓))
10	6, 9	imsubd-P3.28c 271	. . 3 ⊢ (𝛾 → ((∀𝑥𝜑 → ∃𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)))
11		qimeqallhalf-P5 609	. . . 4 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
12	11	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓)))
13	3, 10, 12	dsyl-P3.25 261	. 2 ⊢ (𝛾 → (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
14		dfnfreealt-P6 683	. . 3 ⊢ (Ⅎ𝑥(𝜑 → 𝜓) ↔ (∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)))
15	14	bisym-P3.33b.RC 299	. 2 ⊢ ((∃𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → 𝜓)) ↔ Ⅎ𝑥(𝜑 → 𝜓))
16	13, 15	subimr2-P4.RC 543	1 ⊢ (𝛾 → 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:  nfrandd-P6  816  nfrord-P6  817  nfrbid-P6  818  ndnfrim-P7.3  828