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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 2544 and sbiev 2330 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 2093 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sbievw.is | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | equsalvw 2010 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
4 | 1, 3 | bitri 277 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 |
This theorem is referenced by: sbiedvw 2104 2sbievw 2105 sbievw2 2107 sbid2vw 2260 2mos 2734 cbvabv 2889 clelsb3vOLD 2941 sbralie 3471 sbcco2 3799 sbcie2g 3811 sn-elabg 39153 2reu8i 43361 ichcircshi 43661 |
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