Theorem bj-cbval2vv 36736

Description: Version of cbval2vv 2416 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-cbval2vv.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bj-cbval2vv	⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

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

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

Proof of Theorem bj-cbval2vv

Step	Hyp	Ref	Expression
1		nfv 1913	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1913	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1913	. 2 ⊢ Ⅎ𝑦𝜓
5		bj-cbval2vv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbval2v 2343	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783

This theorem is referenced by: (None)