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Mirrors > Home > MPE Home > Th. List > sb8 | Structured version Visualization version GIF version |
Description: Substitution of variable in universal quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. For a version requiring disjoint variables, but fewer axioms, see sb8f 2350. (Contributed by NM, 16-May-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb8.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb8 | ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb8.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfs1 2491 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
3 | sbequ12 2244 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbval 2397 | 1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 Ⅎwnf 1786 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 |
This theorem is referenced by: sbhb 2524 sb8iota 6465 wl-sb8eut 36061 |
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