Theorem nfbidOLD 2230

Description: Obsolete proof of nfbid 1820 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfbidOLD.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfbidOLD.2	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
nfbidOLD	⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒))

Proof of Theorem nfbidOLD

Step	Hyp	Ref	Expression
1		dfbi2 658	. 2 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
2		nfbidOLD.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		nfbidOLD.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4	2, 3	nfimdOLD 2214	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))
5	3, 2	nfimdOLD 2214	. . 3 ⊢ (𝜑 → Ⅎ𝑥(𝜒 → 𝜓))
6	4, 5	nfandOLD 2220	. 2 ⊢ (𝜑 → Ⅎ𝑥((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)))
7	1, 6	nfxfrdOLD 1826	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ℲwnfOLD 1700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-nfOLD 1712

This theorem is referenced by:  nfbiOLD  2231