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Theorem ax12a2-o 33052
Description: Derive ax-c15 32991 from a hypothesis in the form of ax-12 2031, without using ax-12 2031 or ax-c15 32991. The hypothesis is weaker than ax-12 2031, 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-12 2031, if we also have ax-c11 32989, which this proof uses. As theorem ax12 2287 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 32990 instead of ax-c11 32989. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
ax12a2-o.1 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
Assertion
Ref Expression
ax12a2-o (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ax12a2-o
StepHypRef Expression
1 ax-5 1825 . . 3 (𝜑 → ∀𝑧𝜑)
2 ax12a2-o.1 . . 3 (𝑥 = 𝑧 → (∀𝑧𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
31, 2syl5 33 . 2 (𝑥 = 𝑧 → (𝜑 → ∀𝑥(𝑥 = 𝑧𝜑)))
43ax12v2-o 33051 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-c5 32985  ax-c4 32986  ax-c7 32987  ax-c10 32988  ax-c11 32989  ax-c9 32992
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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