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Theorem ax11a2-o 2202
 Description: Derive ax-11o 2141 from a hypothesis in the form of ax-11 1746, without using ax-11 1746 or ax-11o 2141. The hypothesis is even weaker than ax-11 1746, with z both distinct from x and not occurring in φ. Thus, the hypothesis provides an alternate axiom that can be used in place of ax-11 1746, if we also hvae ax-10o 2139 which this proof uses . As theorem ax11 2155 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-10 2140 instead of ax-10o 2139. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax11a2-o.1 (x = z → (zφx(x = zφ)))
Assertion
Ref Expression
ax11a2-o 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-o
StepHypRef Expression
1 ax-17 1616 . . 3 (φzφ)
2 ax11a2-o.1 . . 3 (x = z → (zφx(x = zφ)))
31, 2syl5 28 . 2 (x = z → (φx(x = zφ)))
43ax11v2-o 2201 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-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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142 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: (None)
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