Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  drnf1 Structured version   GIF version

Theorem drnf1 1599
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
drex2.1 (x x = y → (φψ))
Assertion
Ref Expression
drnf1 (x x = y → (Ⅎxφ ↔ Ⅎyψ))

Proof of Theorem drnf1
StepHypRef Expression
1 drex2.1 . . . 4 (x x = y → (φψ))
21dral1 1596 . . . 4 (x x = y → (xφyψ))
31, 2imbi12d 223 . . 3 (x x = y → ((φxφ) ↔ (ψyψ)))
43dral1 1596 . 2 (x x = y → (x(φxφ) ↔ y(ψyψ)))
5 df-nf 1326 . 2 (Ⅎxφx(φxφ))
6 df-nf 1326 . 2 (Ⅎyψy(ψyψ))
74, 5, 63bitr4g 212 1 (x x = y → (Ⅎxφ ↔ Ⅎyψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1224  Ⅎwnf 1325 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405 This theorem depends on definitions:  df-bi 110  df-nf 1326 This theorem is referenced by:  drnfc1  2172
 Copyright terms: Public domain W3C validator