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Mirrors > Home > MPE Home > Th. List > sb6 | Structured version Visualization version GIF version |
Description: Alternate definition of substitution when variables are disjoint. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left" also holds without a disjoint variable condition (sb2 2500). Theorem sb6f 2533 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2495 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) Revise df-sb 2066. (Revised by BJ, 22-Dec-2020.) Remove use of ax-11 2157. (Revised by Steven Nguyen, 7-Jul-2023.) (Proof shortened by Wolf Lammen, 16-Jul-2023.) |
Ref | Expression |
---|---|
sb6 | ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2066 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | equequ2 2029 | . . . . 5 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡)) | |
3 | 2 | imbi1d 344 | . . . 4 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑))) |
4 | 3 | albidv 1917 | . . 3 ⊢ (𝑦 = 𝑡 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))) |
5 | 4 | equsalvw 2006 | . 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
6 | 1, 5 | bitri 277 | 1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 [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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 |
This theorem is referenced by: 2sb6 2090 sb1v 2091 sb4vOLD 2092 sb2vOLD 2093 sbrimvlem 2097 sbievw 2099 sbcom3vv 2102 nfs1v 2156 sb4av 2240 sb6a 2255 sb56 2273 sb5OLD 2277 sbnvOLD 2318 sbiev 2326 sbequivvOLD 2330 nfsbvOLD 2346 2eu6 2740 nfabdw 3000 iota4 6330 bj-ax12ssb 33986 bj-hbs1 34129 bj-hbsb2av 34131 bj-sbievw1 34164 bj-sbievw2 34165 bj-sbievw 34166 wl-lem-moexsb 34798 wl-dfralsb 34831 wl-dfrabsb 34855 absnsb 43256 |
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