Metamath Proof Explorer


Theorem spimvw

Description: A weak form of specialization. Lemma 8 of KalishMontague p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv and spimfv . (Contributed by NM, 9-Apr-2017)

Ref Expression
Hypothesis spimvw.1 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
Assertion spimvw ( ∀ 𝑥 𝜑𝜓 )

Proof

Step Hyp Ref Expression
1 spimvw.1 ( 𝑥 = 𝑦 → ( 𝜑𝜓 ) )
2 ax-5 ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
3 2 1 spimw ( ∀ 𝑥 𝜑𝜓 )