Theorem cbvex2vw 2068

Description: Rule used to change bound variables, using implicit substitution. Version of cbvex2vv 2452 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 26-Jul-1995.) Avoid ax-13 2410. (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 2066	. 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
3	2	cbvexvw 2064	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  cbvex4vw  2069  cbvopabv  5185  dm0rn0  5912  cbvoprab12v  7498  cbvoprab123vw  36636  cbvoprab23vw  36637  bj-cbvex4vv  37325  funop1  47902  cycldlenngric  48575  uspgrsprf1  48794