Theorem cbval2v 2285

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.)

Hypothesis

Ref	Expression
cbval2v.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbval2v	⊢ (∀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 cbval2v

Step	Hyp	Ref	Expression
1		cbval2v.1	. . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
2	1	cbvaldva 2281	. 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓))
3	2	cbvalv 2273	1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by:  seqf1o  12842  fi1uzind  13279  brfi1indALT  13282  fi1uzindOLD  13285  brfi1indALTOLD  13288  mbfresfi  33456