Theorem bj-cbv3v2 31914

Description: Version of cbv3 2253 with two dv conditions, which does not require ax-11 2021 nor ax-13 2234. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbv3v2.nf	⊢ Ⅎ𝑥𝜓
bj-cbv3v2.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-cbv3v2	⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

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

Proof of Theorem bj-cbv3v2

Step	Hyp	Ref	Expression
1		nfv 1830	. 2 ⊢ Ⅎ𝑦∀𝑥𝜑
2		bj-cbv3v2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
3		bj-cbv3v2.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimv1 2101	. 2 ⊢ (∀𝑥𝜑 → 𝜓)
5	1, 4	alrimi 2069	1 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-cbv3hv2  31915