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Mirrors > Home > MPE Home > Th. List > sbcied | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.) Avoid ax-10 2138, ax-12 2172. (Revised by Gino Giotto, 12-Oct-2024.) |
Ref | Expression |
---|---|
sbcied.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
sbcied.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcied | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sbc 3779 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
2 | sbcied.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | sbcied.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | elabd3 3662 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒)) |
5 | 1, 4 | bitrid 283 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 [wsbc 3778 |
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 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3779 |
This theorem is referenced by: sbcied2 3825 sbc2ie 3861 sbc2iedv 3863 sbc3ie 3864 sbcralt 3867 csbied 3932 euotd 5514 fmptsnd 7167 riota5f 7394 fpwwe2lem11 10636 fpwwe2lem12 10637 brfi1uzind 14459 opfi1uzind 14462 sbcie3s 17095 issubc 17785 gsumvalx 18595 dmdprd 19868 dprdval 19873 issrg 20011 issrng 20458 islmhm 20638 isphl 21181 istmd 23578 istgp 23581 isnlm 24192 isclm 24580 iscph 24687 iscms 24862 limcfval 25389 ewlksfval 28858 sbcies 31728 abfmpeld 31879 abfmpel 31880 isomnd 32219 isorng 32417 rprmval 32633 f1o2d2 41055 |
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