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Theorem ax12a2-o 33253
 Description: Derive ax-c15 33192 from a hypothesis in the form of ax-12 2034, without using ax-12 2034 or ax-c15 33192. The hypothesis is weaker than ax-12 2034, 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 2034, if we also have ax-c11 33190, which this proof uses. As theorem ax12 2292 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n 33191 instead of ax-c11 33190. (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 1827 . . 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 33252 1 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-c5 33186  ax-c4 33187  ax-c7 33188  ax-c10 33189  ax-c11 33190  ax-c9 33193 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
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