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Mirrors > Home > ILE Home > Th. List > sbh | GIF version |
Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 17-Oct-2004.) |
Ref | Expression |
---|---|
sbh.1 | ⊢ (φ → ∀xφ) |
Ref | Expression |
---|---|
sbh | ⊢ ([y / x]φ ↔ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb1 1646 | . . . 4 ⊢ ([y / x]φ → ∃x(x = y ∧ φ)) | |
2 | sbh.1 | . . . . 5 ⊢ (φ → ∀xφ) | |
3 | 2 | 19.41h 1572 | . . . 4 ⊢ (∃x(x = y ∧ φ) ↔ (∃x x = y ∧ φ)) |
4 | 1, 3 | sylib 127 | . . 3 ⊢ ([y / x]φ → (∃x x = y ∧ φ)) |
5 | 4 | simprd 107 | . 2 ⊢ ([y / x]φ → φ) |
6 | stdpc4 1655 | . . 3 ⊢ (∀xφ → [y / x]φ) | |
7 | 2, 6 | syl 14 | . 2 ⊢ (φ → [y / x]φ) |
8 | 5, 7 | impbii 117 | 1 ⊢ ([y / x]φ ↔ 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-4 1397 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbf 1657 sb6x 1659 nfs1f 1660 hbs1f 1661 sbid2h 1726 sblimv 1771 sbrim 1827 sbrbif 1833 elsb3 1849 elsb4 1850 |
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