Theorem spsv 1987

Description: Generalization of antecedent. A trivial weak version of sps 2186 avoiding ax-12 2178. (Contributed by SN, 13-Nov-2025.) (Proof shortened by WL, 19-Nov-2025.)

Hypothesis

Ref	Expression
spsv.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
spsv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem spsv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		spsv.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	a1i 11	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	2	spimvw 1986	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  sbcal  3830