Description: Show that the original axiom ax-c10 can be derived from ax6 and axc7 (on top of propositional calculus, ax-gen , and ax-4 ). See ax6fromc10 for the rederivation of ax6 from ax-c10 .
Normally, axc10 should be used rather than ax-c10 , except by theorems specifically studying the latter's properties. Usage of this theorem has been discouraged later on to avoid ax-13 propagation. Check out bj-axc10v for a weaker version requiring fewer axioms. (Contributed by NM, 5-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | axc10 | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → 𝜑 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax6 | ⊢ ¬ ∀ 𝑥 ¬ 𝑥 = 𝑦 | |
| 2 | con3 | ⊢ ( ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → ( ¬ ∀ 𝑥 𝜑 → ¬ 𝑥 = 𝑦 ) ) | |
| 3 | 2 | al2imi | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝑥 = 𝑦 ) ) |
| 4 | 1 3 | mtoi | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 ) |
| 5 | axc7 | ⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 ) | |
| 6 | 4 5 | syl | ⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → ∀ 𝑥 𝜑 ) → 𝜑 ) |