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| Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) | 
| Ref | Expression | 
|---|---|
| sbceqbid.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| sbceqbid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | 
| Ref | Expression | 
|---|---|
| sbceqbid | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbceqbid.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | sbceqbid.2 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 3 | 2 | abbidv 2807 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) | 
| 4 | 1, 3 | eleq12d 2834 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})) | 
| 5 | df-sbc 3788 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
| 6 | df-sbc 3788 | . 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}) | |
| 7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {cab 2713 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: sbcbidv 3844 frpoins3xpg 8166 frpoins3xp3g 8167 fpwwe2cbv 10671 fpwwe2lem2 10673 fpwwe2lem3 10674 fi1uzind 14547 isprs 18343 isdrs 18348 istos 18464 isdlat 18568 issrg 20186 islmod 20863 fdc 37753 hdmap1ffval 41798 hdmap1fval 41799 hdmapffval 41829 hdmapfval 41830 hgmapffval 41888 hgmapfval 41889 sbccomieg 42809 rexrabdioph 42810 | 
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