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Theorem drnf1 1618
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 1615 . . . 4 (x x = y → (xφyψ))
31, 2imbi12d 223 . . 3 (x x = y → ((φxφ) ↔ (ψyψ)))
43dral1 1615 . 2 (x x = y → (x(φxφ) ↔ y(ψyψ)))
5 df-nf 1347 . 2 (Ⅎxφx(φxφ))
6 df-nf 1347 . 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 1240  wnf 1346
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-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
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  drnfc1  2191
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