Theorem drnfc1 2997

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-11 2161. (Revised by Wolf Lammen, 10-May-2023.) (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	1	eleq2d 2898	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	2	drnf1 2465	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵))
4	3	albidv 1921	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵))
5		df-nfc 2963	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴)
6		df-nfc 2963	. 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)
7	4, 5, 6	3bitr4g 316	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-13 2390  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by:  nfabd2  3002  nfabd2OLD  3003  nfcvb  5277  nfriotad  7125  bj-nfcsym  34218