Metamath Proof Explorer


Theorem ax12

Description: Rederivation of Axiom ax-12 from ax12v (used only via sp ) , axc11r , and axc15 (on top of Tarski's FOL). Since this version depends on ax-13 , usage of the weaker ax12v , ax12w , ax12i are preferred. (Contributed by NM, 22-Jan-2007) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020) (Proof shortened by BJ, 4-Jul-2021) (New usage is discouraged.)

Ref Expression
Assertion ax12 ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 axc11r ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 𝜑 ) )
2 ala1 ( ∀ 𝑥 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) )
3 1 2 syl6 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )
4 3 a1d ( ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ) )
5 sp ( ∀ 𝑦 𝜑𝜑 )
6 axc15 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ) )
7 5 6 syl7 ( ¬ ∀ 𝑥 𝑥 = 𝑦 → ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) ) )
8 4 7 pm2.61i ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦𝜑 ) ) )