Theorem nfbidfOLD 2189

Description: Obsolete proof of nfbidf 2079 as of 6-Oct-2021. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfbidfOLD.1	⊢ Ⅎ𝑥𝜑
nfbidfOLD.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
nfbidfOLD	⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Proof of Theorem nfbidfOLD

Step	Hyp	Ref	Expression
1		nfbidfOLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfbidfOLD.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	1, 2	albidOLD 2187	. . . 4 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
4	2, 3	imbi12d 333	. . 3 ⊢ (𝜑 → ((𝜓 → ∀𝑥𝜓) ↔ (𝜒 → ∀𝑥𝜒)))
5	1, 4	albidOLD 2187	. 2 ⊢ (𝜑 → (∀𝑥(𝜓 → ∀𝑥𝜓) ↔ ∀𝑥(𝜒 → ∀𝑥𝜒)))
6		df-nfOLD 1712	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
7		df-nfOLD 1712	. 2 ⊢ (Ⅎ𝑥𝜒 ↔ ∀𝑥(𝜒 → ∀𝑥𝜒))
8	5, 6, 7	3bitr4g 302	1 ⊢ (𝜑 → (Ⅎ𝑥𝜓 ↔ Ⅎ𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  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-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nfOLD 1712

This theorem is referenced by: (None)