Theorem nfrand-P6 690

Description: ENF Over Conjunction.

Hypotheses

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

Assertion

Ref	Expression
nfrand-P6	⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Proof of Theorem nfrand-P6

Step	Hyp	Ref	Expression
1		nfrand-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfrand-P6.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
3		nfrneg-P6 688	. . . . 5 ⊢ (Ⅎ𝑥 ¬ 𝜓 ↔ Ⅎ𝑥𝜓)
4	2, 3	bimpr-P4.RC 534	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜓
5	1, 4	nfrim-P6 689	. . 3 ⊢ Ⅎ𝑥(𝜑 → ¬ 𝜓)
6		nfrneg-P6 688	. . 3 ⊢ (Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓) ↔ Ⅎ𝑥(𝜑 → ¬ 𝜓))
7	5, 6	bimpr-P4.RC 534	. 2 ⊢ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓)
8		andasim-P3.46a 356	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (𝜑 → ¬ 𝜓))
9	8	nfrleq-P6 687	. 2 ⊢ (Ⅎ𝑥(𝜑 ∧ 𝜓) ↔ Ⅎ𝑥 ¬ (𝜑 → ¬ 𝜓))
10	7, 9	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:  ¬ wff-neg 9   → wff-imp 10   ∧ wff-and 132  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:  nfrbi-P6  691  nfradd-P6  781  nfrmult-P6  782