Theorem bj-cbvex2vv 37714

Description: Version of cbvex2vv 2443 with a disjoint variable condition, which does not require ax-13 2401. (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 1946	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1946	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1946	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1946	. 2 ⊢ Ⅎ𝑦𝜓
5		bj-cbval2vv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvex2v 2373	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816

This theorem is used by: (None)