Theorem bj-cbval2vv 34614

Description: Version of cbval2vv 2413 with a disjoint variable condition, which does not require ax-13 2372. (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 1921	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1921	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1921	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1921	. 2 ⊢ Ⅎ𝑦𝜓
5		bj-cbval2vv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbval2v 2345	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1540

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 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)