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Mirrors > Home > MPE Home > Th. List > sb7f | Structured version Visualization version GIF version |
Description: This version of dfsb7 2443 does not require that 𝜑 and 𝑧 be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-5 1827 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1868 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.) |
Ref | Expression |
---|---|
sb7f.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sb7f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb7f.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | sb5f 2374 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) |
3 | 2 | sbbii 1874 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑)) |
4 | 1 | sbco2 2403 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
5 | sb5 2418 | . 2 ⊢ ([𝑦 / 𝑧]∃𝑥(𝑥 = 𝑧 ∧ 𝜑) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) | |
6 | 3, 4, 5 | 3bitr3i 289 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 Ⅎwnf 1699 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 |
This theorem is referenced by: sb7h 2442 dfsb7 2443 |
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