Theorem sbalexi 42705

Description: Inference form of sbalex 2254, avoiding ax-10 2152 by using ax-gen 1802. (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 2192	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → (𝑥 = 𝑦 → 𝜑))
3	1, 2	ax-mp 5	. 2 ⊢ (𝑥 = 𝑦 → 𝜑)
4	3	ax-gen 1802	1 ⊢ ∀𝑥(𝑥 = 𝑦 → 𝜑)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by: (None)