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Mirrors > Home > MPE Home > Th. List > cbvaldva | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.) Remove dependency on ax-10 2019. (Revised by Wolf Lammen, 18-Jul-2021.) |
Ref | Expression |
---|---|
cbvaldva.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
cbvaldva | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaldva.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | expcom 451 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
3 | 2 | pm5.74d 262 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
4 | 3 | cbvalv 2273 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜑 → 𝜒)) |
5 | 19.21v 1868 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)) | |
6 | 19.21v 1868 | . . 3 ⊢ (∀𝑦(𝜑 → 𝜒) ↔ (𝜑 → ∀𝑦𝜒)) | |
7 | 4, 5, 6 | 3bitr3i 290 | . 2 ⊢ ((𝜑 → ∀𝑥𝜓) ↔ (𝜑 → ∀𝑦𝜒)) |
8 | 7 | pm5.74ri 261 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒)) |
Colors of variables: wff setvar class |
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: cbvexdva 2283 cbval2v 2285 cbvraldva2 3175 |
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