Theorem bj-cbvaldvav 37157

Description: Version of cbvaldva 2417 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-cbvaldvav.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-cbvaldvav	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

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

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

Proof of Theorem bj-cbvaldvav

Step	Hyp	Ref	Expression
1		nfv 1921	. 2 ⊢ Ⅎ𝑦𝜑
2		nfvd 1922	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜓)
3		bj-cbvaldvav.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	3	ex 413	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
5	1, 2, 4	bj-cbvaldv 37153	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)