Theorem cbval2vw 2070

Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2445 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) Avoid ax-13 2404. (Revised by GG, 10-Jan-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbval2vw	⊢ (∀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 cbval2vw

Step	Hyp	Ref	Expression
1		cbval2vw.1	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
2	1	cbvaldvaw 2068	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
3	2	cbvalvw 2066	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  seqf1o  14075  fi1uzind  14540  brfi1indALT  14543  mh-unprimbi  37075  mbfresfi  38337