Theorem drnfc1 2916

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2114, ax-11 2160. (Revised by Wolf Lammen, 22-Sep-2024.) (New usage is discouraged.)

Hypothesis

Ref	Expression
drnfc1.1	⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)

Assertion

Ref	Expression
drnfc1	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

Proof of Theorem drnfc1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		drnfc1.1	. . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)
2		eleq2w2 2732	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
4	3	drnf1 2442	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵))
5	4	albidv 1928	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵))
6		df-nfc 2879	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴)
7		df-nfc 2879	. 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)
8	5, 6, 7	3bitr4g 317	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   = wceq 1543  Ⅎwnf 1791   ∈ wcel 2112  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-10 2143  ax-12 2177  ax-13 2371  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-cleq 2728  df-nfc 2879

This theorem is referenced by:  nfabd2  2923  nfcvb  5254  nfriotad  7160  bj-nfcsym  34771