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 = 𝑧𝜑 ) ) )
Assertion ax12a2-o ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ) )

Proof

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