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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 2802 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
4 | 1, 3 | eleq12d 2828 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})) |
5 | df-sbc 3779 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
6 | df-sbc 3779 | . 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 205 = 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-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3779 |
This theorem is referenced by: sbcbidv 3837 frpoins3xpg 8126 frpoins3xp3g 8127 fpwwe2cbv 10625 fpwwe2lem2 10627 fpwwe2lem3 10628 fi1uzind 14458 isprs 18250 isdrs 18254 istos 18371 isdlat 18475 issrg 20011 islmod 20475 fdc 36613 hdmap1ffval 40666 hdmap1fval 40667 hdmapffval 40697 hdmapfval 40698 hgmapffval 40756 hgmapfval 40757 sbccomieg 41531 rexrabdioph 41532 |
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