Theorem sbalexi 43359

Description: Inference form of sbalex 2278, avoiding ax-10 2177 by using ax-gen 1827. (Contributed by SN, 12-Aug-2025.)

Hypothesis

Ref	Expression
sbalexi.1	⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)

Assertion

Ref	Expression
sbalexi	⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem sbalexi

Step	Hyp	Ref	Expression
1		sbalexi.1	. . 3 ⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
2		ax12ev2 2215	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
3	1, 2	ax-mp 5	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
4	3	ax-gen 1827	1 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃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: (None)