Theorem nfceqdf 2937

Description: An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)

Hypotheses

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

Assertion

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

Proof of Theorem nfceqdf

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 2864	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵))
4	1, 3	nfbidf 2259	. . 3 ⊢ (𝜑 → (Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ Ⅎ𝑥 𝑦 ∈ 𝐵))
5	4	albidv 2016	. 2 ⊢ (𝜑 → (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵))
6		df-nfc 2930	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
7		df-nfc 2930	. 2 ⊢ (Ⅎ𝑥𝐵 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐵)
8	5, 6, 7	3bitr4g 306	1 ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1651   = wceq 1653  Ⅎwnf 1879   ∈ wcel 2157  Ⅎwnfc 2928

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876  df-nf 1880  df-cleq 2792  df-clel 2795  df-nfc 2930

This theorem is referenced by:  nfceqi  2938  nfopd  4610  dfnfc2  4648  nfimad  5692  nffvd  6423  riotasv2d  34978  nfcxfrdf  34987  nfded  34988  nfded2  34989  nfunidALT2  34990