Theorem drnfc1 2925

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2110, ax-11 2156. (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 2734	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
4	3	drnf1 2443	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵))
5	4	albidv 1924	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵))
6		df-nfc 2888	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴)
7		df-nfc 2888	. 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)
8	5, 6, 7	3bitr4g 313	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2118  ax-10 2139  ax-12 2173  ax-13 2372  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788  df-cleq 2730  df-nfc 2888

This theorem is referenced by:  nfabd2  2932  nfcvb  5294  nfriotad  7224  bj-nfcsym  35011