Theorem cbvalivw 2111

Description: Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
cbvalivw.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
cbvalivw	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvalivw

Step	Hyp	Ref	Expression
1		cbvalivw.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
2	1	spimvw 2103	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
3	2	alrimiv 2026	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075

This theorem depends on definitions:  df-bi 199  df-ex 1879

This theorem is referenced by:  alcomiw  2145  alcomiwOLD  2146  cbvaev  2153  wl-cbvmotv  33840  axc11next  39445