Theorem nfrord-P6 817

Description: ENF Over Disjunction (deductive form).

Hypotheses

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

Assertion

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

Proof of Theorem nfrord-P6

Step	Hyp	Ref	Expression
1		nfrord-P6.1	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜑)
2		nfrneg-P6 688	. . . . 5 ⊢ (Ⅎ𝑥 ¬ 𝜑 ↔ Ⅎ𝑥𝜑)
3	2	bisym-P3.33b.RC 299	. . . 4 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
4	1, 3	subimr2-P4.RC 543	. . 3 ⊢ (𝛾 → Ⅎ𝑥 ¬ 𝜑)
5		nfrord-P6.2	. . 3 ⊢ (𝛾 → Ⅎ𝑥𝜓)
6	4, 5	nfrimd-P6 815	. 2 ⊢ (𝛾 → Ⅎ𝑥(¬ 𝜑 → 𝜓))
7		orasim-P3.48a 361	. . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))
8	7	nfrleq-P6 687	. . 3 ⊢ (Ⅎ𝑥(𝜑 ∨ 𝜓) ↔ Ⅎ𝑥(¬ 𝜑 → 𝜓))
9	8	bisym-P3.33b.RC 299	. 2 ⊢ (Ⅎ𝑥(¬ 𝜑 → 𝜓) ↔ Ⅎ𝑥(𝜑 ∨ 𝜓))
10	6, 9	subimr2-P4.RC 543	1 ⊢ (𝛾 → Ⅎ𝑥(𝜑 ∨ 𝜓))

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∨ wff-or 144  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:  ndnfror-P7.5  830