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Theorem dral1 1965
 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 1953 . . . 4 (x x = yxx x = y)
2 dral1.1 . . . . 5 (x x = y → (φψ))
32biimpd 198 . . . 4 (x x = y → (φψ))
41, 3alimdh 1563 . . 3 (x x = y → (xφxψ))
5 ax10o 1952 . . 3 (x x = y → (xψyψ))
64, 5syld 40 . 2 (x x = y → (xφyψ))
7 hbae 1953 . . . 4 (x x = yyx x = y)
82biimprd 214 . . . 4 (x x = y → (ψφ))
97, 8alimdh 1563 . . 3 (x x = y → (yψyφ))
10 ax10o 1952 . . . 4 (y y = x → (yφxφ))
1110aecoms 1947 . . 3 (x x = y → (yφxφ))
129, 11syld 40 . 2 (x x = y → (yψxφ))
136, 12impbid 183 1 (x x = y → (xφyψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  drex1  1967  drnf1  1969  equveli  1988  a16gALT  2049  sb9i  2094  ralcom2  2775
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