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Metamath Proof Explorer

Theorem ax12a2-o

Description: Derive ax-c15 from a hypothesis in the form of ax-12 , without using ax-12 or ax-c15 . The hypothesis is weaker than ax-12 , with z both distinct from x and not occurring in ph . Thus, the hypothesis provides an alternate axiom that can be used in place of ax-12 , if we also have ax-c11 , which this proof uses. As Theorem ax12 shows, the distinct variable conditions are optional. An open problem is whether we can derive this with ax-c11n instead of ax-c11 . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypothesis ax12a2-o.1 ⊢ x = z → ∀ z φ → ∀ x x = z → φ
Assertion ax12a2-o ⊢ ¬ ∀ x x = y → x = y → φ → ∀ x x = y → φ

Proof

Step Hyp Ref Expression
1 ax12a2-o.1 ⊢ x = z → ∀ z φ → ∀ x x = z → φ
2 ax-5 ⊢ φ → ∀ z φ
3 2 1 syl5 ⊢ x = z → φ → ∀ x x = z → φ
4 3 ax12v2-o ⊢ ¬ ∀ x x = y → x = y → φ → ∀ x x = y → φ