Theorem nfbiiOLD 1836

Description: Obsolete proof of nfbii 1778 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
nfbiiOLD.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
nfbiiOLD	⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Proof of Theorem nfbiiOLD

Step	Hyp	Ref	Expression
1		nfbiiOLD.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
2	1	albii 1747	. . . 4 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
3	1, 2	imbi12i 340	. . 3 ⊢ ((𝜑 → ∀𝑥𝜑) ↔ (𝜓 → ∀𝑥𝜓))
4	3	albii 1747	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5		df-nfOLD 1721	. 2 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
6		df-nfOLD 1721	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
7	4, 5, 6	3bitr4i 292	1 ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ℲwnfOLD 1709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-nfOLD 1721

This theorem is referenced by:  nfxfrOLD  1837  nfxfrdOLD  1838