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Theorem drnf2 1600
 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
drnf2 (x x = y → (Ⅎzφ ↔ Ⅎzψ))

Proof of Theorem drnf2
StepHypRef Expression
1 drex2.1 . . . 4 (x x = y → (φψ))
21dral2 1597 . . . 4 (x x = y → (zφzψ))
31, 2imbi12d 223 . . 3 (x x = y → ((φzφ) ↔ (ψzψ)))
43dral2 1597 . 2 (x x = y → (z(φzφ) ↔ z(ψzψ)))
5 df-nf 1326 . 2 (Ⅎzφz(φzφ))
6 df-nf 1326 . 2 (Ⅎzψz(ψzψ))
74, 5, 63bitr4g 212 1 (x x = y → (Ⅎzφ ↔ Ⅎzψ))
 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:  nfsbxy  1796  nfsbxyt  1797  drnfc2  2173
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