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Mirrors > Home > ILE Home > Th. List > sbcie2g | GIF version |
Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 2943 avoids a disjointness condition on 𝑥 and 𝐴 by substituting twice. (Contributed by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
sbcie2g.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
sbcie2g.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbcie2g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2911 | . 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | sbcie2g.2 | . 2 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
3 | sbsbc 2913 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
5 | sbcie2g.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | sbie 1764 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
7 | 3, 6 | bitr3i 185 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
8 | 1, 2, 7 | vtoclbg 2747 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 [wsb 1735 [wsbc 2909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 |
This theorem is referenced by: sbcel2gv 2972 csbie2g 3050 brab1 3975 |
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