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Mirrors > Home > MPE Home > Th. List > sbceqbid | Structured version Visualization version GIF version |
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 2806 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
4 | 1, 3 | eleq12d 2833 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})) |
5 | df-sbc 3792 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
6 | df-sbc 3792 | . 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 1537 ∈ wcel 2106 {cab 2712 [wsbc 3791 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: sbcbidv 3851 frpoins3xpg 8164 frpoins3xp3g 8165 fpwwe2cbv 10668 fpwwe2lem2 10670 fpwwe2lem3 10671 fi1uzind 14543 isprs 18354 isdrs 18359 istos 18476 isdlat 18580 issrg 20206 islmod 20879 fdc 37732 hdmap1ffval 41778 hdmap1fval 41779 hdmapffval 41809 hdmapfval 41810 hgmapffval 41868 hgmapfval 41869 sbccomieg 42781 rexrabdioph 42782 |
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