Theorem cbvex2vw 2074

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2449 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid ax-13 2407. (Revised by GG, 10-Jan-2024.)

Hypothesis

Ref	Expression
cbval2vw.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvex2vw	⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

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

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

Proof of Theorem cbvex2vw

Step	Hyp	Ref	Expression
1		cbval2vw.1	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
2	1	cbvexdvaw 2072	. 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
3	2	cbvexvw 2070	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  cbvex4vw  2075  cbvopabv  5189  dm0rn0  5919  cbvoprab12v  7513  cbvoprab123vw  36792  cbvoprab23vw  36793  bj-cbvex4vv  37481  funop1  48061  cycldlenngric  48734  uspgrsprf1  48953