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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 1777 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
2 | sbh.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
3 | 2 | 19.41h 1696 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜑)) |
4 | 1, 3 | sylib 122 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 → (∃𝑥 𝑥 = 𝑦 ∧ 𝜑)) |
5 | 4 | simprd 114 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑) |
6 | stdpc4 1786 | . . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
7 | 2, 6 | syl 14 | . 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑) |
8 | 5, 7 | impbii 126 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 ∃wex 1503 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-sb 1774 |
This theorem is referenced by: sbf 1788 sb6x 1790 nfs1f 1791 hbs1f 1792 sbid2h 1860 sblimv 1906 sbrim 1968 sbrbif 1974 elsb1 2167 elsb2 2168 |
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