Theorem cbvex2vv 2445

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2403. Use the weaker cbvex2vw 2070 if possible. (Contributed by NM, 26-Jul-1995.) Remove dependency on ax-10 2175. (Revised by Wolf Lammen, 18-Jul-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
cbval2vv.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvex2vv	⊢ (∃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 cbvex2vv

Step	Hyp	Ref	Expression
1		cbval2vv.1	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
2	1	cbvexdva 2441	. 2 ⊢ (𝑥 = 𝑧 → (∃𝑦𝜑 ↔ ∃𝑤𝜓))
3	2	cbvexv 2432	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-11 2191  ax-12 2212  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-nf 1813

This theorem is used by:  cbvex4v  2446