Theorem drnfc2 2335

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
drnfc2	⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Proof of Theorem drnfc2

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 2245	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))
3	2	drnf2 1732	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ Ⅎ𝑧 𝑤 ∈ 𝐵))
4	3	dral2 1729	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵))
5		df-nfc 2306	. 2 ⊢ (Ⅎ𝑧𝐴 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐴)
6		df-nfc 2306	. 2 ⊢ (Ⅎ𝑧𝐵 ↔ ∀𝑤Ⅎ𝑧 𝑤 ∈ 𝐵)
7	4, 5, 6	3bitr4g 223	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351   = wceq 1353  Ⅎwnf 1458   ∈ wcel 2146  Ⅎwnfc 2304

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-cleq 2168  df-clel 2171  df-nfc 2306

This theorem is referenced by: (None)