Theorem cbval2vv 2435

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker cbval2vw 2047 if possible. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2145. (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
cbval2vv	⊢ (∀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 cbval2vv

Step	Hyp	Ref	Expression
1		cbval2vv.1	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
2	1	cbvaldva 2430	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
3	2	cbvalv 2418	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-11 2161  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785

This theorem is referenced by:  brfi1indALT  13859