Theorem drnfc1 2353

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)

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 2263	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	2	drnf1 1744	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ Ⅎ𝑦 𝑤 ∈ 𝐵))
4	3	dral2 1742	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵))
5		df-nfc 2325	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑤Ⅎ𝑥 𝑤 ∈ 𝐴)
6		df-nfc 2325	. 2 ⊢ (Ⅎ𝑦𝐵 ↔ ∀𝑤Ⅎ𝑦 𝑤 ∈ 𝐵)
7	4, 5, 6	3bitr4g 223	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  Ⅎwnf 1471   ∈ wcel 2164  Ⅎwnfc 2323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-cleq 2186  df-clel 2189  df-nfc 2325

This theorem is referenced by: (None)