Theorem drnfc1 2505

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 2420	. . . 4 ⊢ (∀x x = y → (w ∈ A ↔ w ∈ B))
3	2	drnf1 1969	. . 3 ⊢ (∀x x = y → (Ⅎx w ∈ A ↔ Ⅎy w ∈ B))
4	3	dral2 1966	. 2 ⊢ (∀x x = y → (∀wℲx w ∈ A ↔ ∀wℲy w ∈ B))
5		df-nfc 2478	. 2 ⊢ (ℲxA ↔ ∀wℲx w ∈ A)
6		df-nfc 2478	. 2 ⊢ (ℲyB ↔ ∀wℲy w ∈ B)
7	4, 5, 6	3bitr4g 279	1 ⊢ (∀x x = y → (ℲxA ↔ ℲyB))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-nfc 2478

This theorem is referenced by:  nfabd2  2507