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| Mirrors > Home > MPE Home > Th. List > sb8v | Structured version Visualization version GIF version | ||
| Description: Substitution of variable in universal quantifier. Version of sb8f 2356 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2177. (Contributed by SN, 5-Dec-2024.) |
| Ref | Expression |
|---|---|
| sb8v | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sb6 2085 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
| 2 | 1 | albii 1819 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
| 3 | alcom 2159 | . 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)) | |
| 4 | equcom 2017 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
| 5 | 4 | imbi1i 349 | . . . . 5 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑦 = 𝑥 → 𝜑)) |
| 6 | 5 | albii 1819 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑)) |
| 7 | equsv 2002 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑) | |
| 8 | 6, 7 | bitri 275 | . . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) |
| 9 | 8 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥𝜑) |
| 10 | 2, 3, 9 | 3bitrri 298 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 [wsb 2064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-11 2157 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 |
| This theorem is referenced by: sbnf2 2361 sb8eulem 2598 cbvralsvw 3317 abv 3492 abvALT 3493 wl-sb8eutv 37580 sbcalf 38121 |
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