Theorem wl-cbv3vv 33307

Description: Avoiding ax-11 2034. (Contributed by Wolf Lammen, 30-Aug-2021.)

Hypotheses

Ref	Expression
wl-cbv3vv.nf	⊢ Ⅎ𝑥𝜓
wl-cbv3vv.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
wl-cbv3vv	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem wl-cbv3vv

Step	Hyp	Ref	Expression
1		wl-cbv3vv.nf	. . 3 ⊢ Ⅎ𝑥𝜓
2		wl-cbv3vv.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimv1 2115	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
4	3	alrimiv 1855	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1481  Ⅎwnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)