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Theorem dral1 1615
 Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.)
Hypothesis
Ref Expression
dral1.1 (x x = y → (φψ))
Assertion
Ref Expression
dral1 (x x = y → (xφyψ))

Proof of Theorem dral1
StepHypRef Expression
1 hbae 1603 . . . 4 (x x = yxx x = y)
2 dral1.1 . . . . 5 (x x = y → (φψ))
32biimpd 132 . . . 4 (x x = y → (φψ))
41, 3alimdh 1353 . . 3 (x x = y → (xφxψ))
5 ax10o 1600 . . 3 (x x = y → (xψyψ))
64, 5syld 40 . 2 (x x = y → (xφyψ))
7 hbae 1603 . . . 4 (x x = yyx x = y)
82biimprd 147 . . . 4 (x x = y → (ψφ))
97, 8alimdh 1353 . . 3 (x x = y → (yψyφ))
10 ax10o 1600 . . . 4 (y y = x → (yφxφ))
1110alequcoms 1406 . . 3 (x x = y → (yφxφ))
129, 11syld 40 . 2 (x x = y → (yψxφ))
136, 12impbid 120 1 (x x = y → (xφyψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240 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 This theorem is referenced by:  drnf1  1618  equveli  1639  a16g  1741
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