Theorem cbvaliw 2009

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.)

Hypotheses

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

Assertion

Ref	Expression
cbvaliw	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvaliw

Step	Hyp	Ref	Expression
1		cbvaliw.1	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
2		cbvaliw.2	. . 3 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
3		cbvaliw.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimw 1974	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5	1, 4	alrimih 1826	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-6 1971

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  spfw  2036  cbvalw  2038