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Mirrors > Home > MPE Home > Th. List > cbval2vw | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Version of cbval2vv 2408 with more disjoint variable conditions, which requires fewer axioms . (Contributed by NM, 4-Feb-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbval2vw.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbval2vw | ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbval2vw.1 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓)) | |
2 | 1 | cbvaldvaw 2037 | . 2 ⊢ (𝑥 = 𝑧 → (∀𝑦𝜑 ↔ ∀𝑤𝜓)) |
3 | 2 | cbvalvw 2035 | 1 ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑧∀𝑤𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1778 |
This theorem is referenced by: seqf1o 13792 fi1uzind 14239 brfi1indALT 14242 mbfresfi 35851 |
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