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Mirrors > Home > ILE Home > Th. List > sb5 | GIF version |
Description: Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.) |
Ref | Expression |
---|---|
sb5 | ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1763 | . 2 ⊢ ([y / x]φ ↔ ∀x(x = y → φ)) | |
2 | sb56 1762 | . 2 ⊢ (∃x(x = y ∧ φ) ↔ ∀x(x = y → φ)) | |
3 | 1, 2 | bitr4i 176 | 1 ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbnv 1765 sborv 1767 sbi2v 1769 nfsbxy 1815 nfsbxyt 1816 2sb5 1856 dfsb7 1864 sb7f 1865 sbexyz 1876 sbc5 2781 |
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