Theorem bj-ax12v3

Description: A weak version of ax-12 which is stronger than ax12v . Note that if one assumes reflexivity of equality |- x = x ( equid ), then bj-ax12v3 implies ax-5 over modal logic K (substitute x for y ). See also bj-ax12v3ALT . (Contributed by BJ, 6-Jul-2021) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ax12v3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ax-5	⊢ ( 𝜑 → ∀ 𝑦 𝜑 )
2		ax12	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	1 2	syl5	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )