Theorem drnfc1 2947

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Usage of this theorem is discouraged because it depends on ax-13 2407. (Contributed by Mario Carneiro, 8-Oct-2016.) Avoid ax-8 2148, ax-11 2195. (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 2762	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	1, 2	syl 18	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
4	3	drnf1 2478	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵))
5	4	albidv 1953	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵))
6		df-nfc 2915	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴)
7		df-nfc 2915	. 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)
8	5, 6, 7	3bitr4g 317	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2913

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2156  ax-10 2179  ax-12 2216  ax-13 2407  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-cleq 2758  df-nfc 2915

This theorem is used by:  nfabd2  2951  nfcvb  5352  nfriotad  7391  bj-nfcsym  37575