Theorem ax12ev2c

Description: A commuted form of ax12ev2 . (Contributed by BTernaryTau, 8-Sep-2026)

	Ref	Expression
Assertion	ax12ev2c	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		equcomi	⊢ ( 𝑥 = 𝑦 → 𝑦 = 𝑥 )
2	1	anim1i	⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → ( 𝑦 = 𝑥 ∧ 𝜑 ) )
3	2	eximi	⊢ ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝜑 ) → ∃ 𝑦 ( 𝑦 = 𝑥 ∧ 𝜑 ) )
4		ax12ev2	⊢ ( ∃ 𝑦 ( 𝑦 = 𝑥 ∧ 𝜑 ) → ( 𝑦 = 𝑥 → 𝜑 ) )
5	3 1 4	syl2imc	⊢ ( 𝑥 = 𝑦 → ( ∃ 𝑦 ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜑 ) )