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Mirrors > Home > MPE Home > Th. List > sb7h | Structured version Visualization version GIF version |
Description: This version of dfsb7 2281 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 1907 i.e. that doesn't have the concept of a variable not occurring in a wff. (dfsb1 2506 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb7h.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
sb7h | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb7h.1 | . . 3 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | nf5i 2146 | . 2 ⊢ Ⅎ𝑧𝜑 |
3 | 2 | sb7f 2564 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑧(𝑧 = 𝑦 ∧ ∃𝑥(𝑥 = 𝑧 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1531 ∃wex 1776 [wsb 2065 |
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 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-11 2156 ax-12 2172 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: (None) |
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