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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 2354 with a disjoint variable condition replacing the nonfree hypothesis Ⅎ𝑦𝜑, not requiring ax-12 2175. (Contributed by SN, 5-Dec-2024.) |
Ref | Expression |
---|---|
sb8v | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2083 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | 1 | albii 1816 | . 2 ⊢ (∀𝑦[𝑦 / 𝑥]𝜑 ↔ ∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | alcom 2157 | . 2 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)) | |
4 | equcom 2015 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥) | |
5 | 4 | imbi1i 349 | . . . . 5 ⊢ ((𝑥 = 𝑦 → 𝜑) ↔ (𝑦 = 𝑥 → 𝜑)) |
6 | 5 | albii 1816 | . . . 4 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → 𝜑)) |
7 | equsv 2000 | . . . 4 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑) | |
8 | 6, 7 | bitri 275 | . . 3 ⊢ (∀𝑦(𝑥 = 𝑦 → 𝜑) ↔ 𝜑) |
9 | 8 | albii 1816 | . 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 1535 [wsb 2062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-11 2155 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 |
This theorem is referenced by: sbnf2 2359 sb8eulem 2596 cbvralsvw 3315 abv 3490 abvALT 3491 wl-sb8eutv 37560 sbcalf 38101 |
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