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Mirrors > Home > MPE Home > Th. List > sbievw | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution. Version of sbie 2502 and sbiev 2309 with more disjoint variable conditions, requiring fewer axioms. (Contributed by NM, 30-Jun-1994.) (Revised by BJ, 18-Jul-2023.) |
Ref | Expression |
---|---|
sbievw.is | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
sbievw | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2089 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sbievw.is | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | equsalvw 2008 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
4 | 1, 3 | bitri 275 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 |
This theorem is referenced by: sbiedvw 2097 2sbievw 2098 sbievw2 2100 sbid2vw 2251 cbvsbv 2360 2mos 2646 eqabbw 2810 sbralieALT 3356 rabrabi 3451 ralab 3688 sbcco2 3805 sbcie2g 3820 csbied 3932 ss2abdv 4061 unabw 4298 notabw 4304 ab0w 4374 ab0orv 4379 elabgw 41408 2reu8i 45821 ichcircshi 46122 |
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