Theorem nfrbi-P6 691

Description: ENF Over Equivalency.

Hypotheses

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

Assertion

Ref	Expression
nfrbi-P6	⊢ Ⅎ𝑥(𝜑 ↔ 𝜓)

Proof of Theorem nfrbi-P6

Step	Hyp	Ref	Expression
1		nfrbi-P6.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfrbi-P6.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfrim-P6 689	. . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
4	2, 1	nfrim-P6 689	. . 3 ⊢ Ⅎ𝑥(𝜓 → 𝜑)
5	3, 4	nfrand-P6 690	. 2 ⊢ Ⅎ𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))
6		dfbi-P3.47 358	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	6	nfrleq-P6 687	. 2 ⊢ (Ⅎ𝑥(𝜑 ↔ 𝜓) ↔ Ⅎ𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
8	5, 7	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓)

Syntax hints:   → wff-imp 10   ↔ wff-bi 104   ∧ 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:  trnsvsubw-P6  710  trnsvsub-P6  763  spliteq-P6  776  splitelof-P6  778