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Theorem ax11a2 1993
 Description: Derive ax-11o 2141 from a hypothesis in the form of ax-11 1746. ax-10 2140 and ax-11 1746 are used by the proof, but not ax-10o 2139 or ax-11o 2141. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)
Hypothesis
Ref Expression
ax11a2.1 (x = z → (zφx(x = zφ)))
Assertion
Ref Expression
ax11a2 x x = y → (x = y → (φx(x = yφ))))
Distinct variable groups:   x,z   y,z   φ,z
Allowed substitution hints:   φ(x,y)

Proof of Theorem ax11a2
StepHypRef Expression
1 ax-17 1616 . . 3 (φzφ)
2 ax11a2.1 . . 3 (x = z → (zφx(x = zφ)))
31, 2syl5 28 . 2 (x = z → (φx(x = zφ)))
43ax11v2 1992 1 x x = y → (x = y → (φx(x = yφ))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540 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:  ax11o  1994
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