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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 ) ) ) ) |
| 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 ) ) ) ) |