Theorem nfrbid-P6 818

Description: ENF Over Biconditional (deductive form).

Hypotheses

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

Assertion

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

Proof of Theorem nfrbid-P6

Step	Hyp	Ref	Expression
1		nfrbid-P6.1	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜑)
2		nfrbid-P6.2	. . . 4 ⊢ (𝛾 → Ⅎ𝑥𝜓)
3	1, 2	nfrimd-P6 815	. . 3 ⊢ (𝛾 → Ⅎ𝑥(𝜑 → 𝜓))
4	2, 1	nfrimd-P6 815	. . 3 ⊢ (𝛾 → Ⅎ𝑥(𝜓 → 𝜑))
5	3, 4	nfrandd-P6 816	. 2 ⊢ (𝛾 → Ⅎ𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
6		dfbi-P3.47 358	. . . 4 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
7	6	nfrleq-P6 687	. . 3 ⊢ (Ⅎ𝑥(𝜑 ↔ 𝜓) ↔ Ⅎ𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)))
8	7	bisym-P3.33b.RC 299	. 2 ⊢ (Ⅎ𝑥((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ Ⅎ𝑥(𝜑 ↔ 𝜓))
9	5, 8	subimr2-P4.RC 543	1 ⊢ (𝛾 → 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:  ndnfrbi-P7.6  831