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Mirrors > Home > MPE Home > Th. List > csb2 | Structured version Visualization version GIF version |
Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.) |
Ref | Expression |
---|---|
csb2 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-csb 3807 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | sbc5 3726 | . . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
3 | 2 | abbii 2859 | . 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)} |
4 | 1, 3 | eqtri 2817 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1520 ∃wex 1759 ∈ wcel 2079 {cab 2773 [wsbc 3701 ⦋csb 3806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-v 3434 df-sbc 3702 df-csb 3807 |
This theorem is referenced by: (None) |
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