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Theorem drnf1 1969
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
dral1.1 (x x = y → (φψ))
Assertion
Ref Expression
drnf1 (x x = y → (Ⅎxφ ↔ Ⅎyψ))

Proof of Theorem drnf1
StepHypRef Expression
1 dral1.1 . . . 4 (x x = y → (φψ))
21dral1 1965 . . . 4 (x x = y → (xφyψ))
31, 2imbi12d 311 . . 3 (x x = y → ((φxφ) ↔ (ψyψ)))
43dral1 1965 . 2 (x x = y → (x(φxφ) ↔ y(ψyψ)))
5 df-nf 1545 . 2 (Ⅎxφx(φxφ))
6 df-nf 1545 . 2 (Ⅎyψy(ψyψ))
74, 5, 63bitr4g 279 1 (x x = y → (Ⅎxφ ↔ Ⅎyψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544 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 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545 This theorem is referenced by:  nfald2  1972  drnfc1  2505
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