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Mirrors > Home > MPE Home > Th. List > sbccsb | Structured version Visualization version GIF version |
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
sbccsb | ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2707 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | sbcbii 3832 | . 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
3 | sbcel2 4410 | . 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) | |
4 | 2, 3 | bitr3i 277 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2098 {cab 2703 [wsbc 3772 ⦋csb 3888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-nul 4318 |
This theorem is referenced by: (None) |
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