Theorem bj-cbvex2vv 34911

Description: Version of cbvex2vv 2414 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-cbvex2vv	⊢ (∃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-cbvex2vv

Step	Hyp	Ref	Expression
1		nfv 1918	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1918	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1918	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1918	. 2 ⊢ Ⅎ𝑦𝜓
5		bj-cbval2vv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvex2v 2344	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)