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Mirrors > Home > MPE Home > Th. List > sb6 | Structured version Visualization version Unicode version |
Description: Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" is sb2 2352 and does not require any dv condition. Theorem sb6f 2385 replaces the dv condition with a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) |
Ref | Expression |
---|---|
sb6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1883 | . . 3 | |
2 | sb56 2150 | . . 3 | |
3 | 1, 2 | sylib 208 | . 2 |
4 | sb2 2352 | . 2 | |
5 | 3, 4 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wex 1704 wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-sb 1881 |
This theorem is referenced by: sb5 2430 2sb6 2444 sb6a 2448 2eu6 2558 |
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