Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  drnfc1 Structured version   GIF version

Theorem drnfc1 2191
 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
drnfc1 (x x = y → (xAyB))

Proof of Theorem drnfc1
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 drnfc1.1 . . . . 5 (x x = yA = B)
21eleq2d 2104 . . . 4 (x x = y → (w Aw B))
32drnf1 1618 . . 3 (x x = y → (Ⅎx w A ↔ Ⅎy w B))
43dral2 1616 . 2 (x x = y → (wx w Awy w B))
5 df-nfc 2164 . 2 (xAwx w A)
6 df-nfc 2164 . 2 (yBwy w B)
74, 5, 63bitr4g 212 1 (x x = y → (xAyB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator