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Mirrors > Home > ILE Home > Th. List > sbccsbg | GIF version |
Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) |
Ref | Expression |
---|---|
sbccsbg | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid 2177 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
2 | 1 | sbcbii 3037 | . 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑) |
3 | sbcel2g 3093 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})) | |
4 | 2, 3 | bitr3id 194 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2160 {cab 2175 [wsbc 2977 ⦋csb 3072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-sbc 2978 df-csb 3073 |
This theorem is referenced by: (None) |
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