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Theorem drnfc2 2506
 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 = yA = B)
Assertion
Ref Expression
drnfc2 (x x = y → (zAzB))

Proof of Theorem drnfc2
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5 (x x = yA = B)
21eleq2d 2420 . . . 4 (x x = y → (w Aw B))
32drnf2 1970 . . 3 (x x = y → (Ⅎz w A ↔ Ⅎz w B))
43dral2 1966 . 2 (x x = y → (wz w Awz w B))
5 df-nfc 2478 . 2 (zAwz w A)
6 df-nfc 2478 . 2 (zBwz w B)
74, 5, 63bitr4g 279 1 (x x = y → (zAzB))
 Colors of variables: wff setvar class 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: (None)
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