Theorem nfceqdfOLD 2895

Description: Obsolete version of nfceqdf 2894 as of 23-Aug-2024. (Contributed by Mario Carneiro, 14-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfceqdf.1	⊢ Ⅎ𝑥𝜑
nfceqdf.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
nfceqdfOLD	⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Proof of Theorem nfceqdfOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfceqdf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		nfceqdf.2	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	nfbidf 2213	. . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵))
5	4	albidv 1916	. 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵))
6		df-nfc 2881	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
7		df-nfc 2881	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
8	5, 6, 7	3bitr4g 314	1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1532   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  Ⅎwnfc 2879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-nf 1779  df-cleq 2720  df-clel 2806  df-nfc 2881

This theorem is referenced by: (None)