Theorem cbvaliw

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of KalishMontague p. 86. (Contributed by NM, 19-Apr-2017)

	Ref	Expression
Hypotheses	cbvaliw.1	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
	cbvaliw.2	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
	cbvaliw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
Assertion	cbvaliw	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvaliw.1	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 ∀ 𝑥 𝜑 )
2		cbvaliw.2	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜓 )
3		cbvaliw.3	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → 𝜓 ) )
4	2 3	spimw	⊢ ( ∀ 𝑥 𝜑 → 𝜓 )
5	1 4	alrimih	⊢ ( ∀ 𝑥 𝜑 → ∀ 𝑦 𝜓 )