Theorem spsv 2020

Description: Generalization of antecedent. A trivial weak version of sps 2224 avoiding ax-12 2216. (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 2019	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  sbcal  3805  vn0  4298  bj-vn0ALT  37765