Theorem ax12ev2c 2216

Description: A commuted form of ax12ev2 2215. (Contributed by BTernaryTau, 8-Sep-2026.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem ax12ev2c

Step	Hyp	Ref	Expression
1		equcomi 2049	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
2	1	anim1i 627	. . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → (𝑦 = 𝑥 ∧ 𝜑))
3	2	eximi 1867	. 2 ⊢ (∃𝑦(𝑥 = 𝑦 ∧ 𝜑) → ∃𝑦(𝑦 = 𝑥 ∧ 𝜑))
4		ax12ev2 2215	. 2 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝜑) → (𝑦 = 𝑥 → 𝜑))
5	3, 1, 4	syl2imc 42	1 ⊢ (𝑥 = 𝑦 → (∃𝑦(𝑥 = 𝑦 ∧ 𝜑) → 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  copsexgw  5459  copsexg  5461  cotsexgw  5462