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Mirrors > Home > ILE Home > Th. List > vtoclbg | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
vtoclbg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
vtoclbg.2 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
vtoclbg.3 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
vtoclbg | ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtoclbg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
2 | vtoclbg.2 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) | |
3 | 1, 2 | bibi12d 233 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
4 | vtoclbg.3 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
5 | 3, 4 | vtoclg 2658 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 |
This theorem is referenced by: pm13.183 2732 sbc8g 2822 sbcco 2836 sbc5 2838 sbcie2g 2847 eqsbc3 2853 sbcng 2854 sbcimg 2855 sbcan 2856 sbcang 2857 sbcor 2858 sbcorg 2859 sbcbig 2860 sbcal 2865 sbcalg 2866 sbcex2 2867 sbcexg 2868 sbcel1v 2876 sbcralg 2892 sbcreug 2894 sbcel12g 2921 sbceqg 2922 csbiebg 2945 elpwg 3390 snssg 3522 preq12bg 3565 elintg 3644 elintrabg 3649 sbcbrg 3834 opelresg 4637 domeng 6256 |
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