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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 234 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
4 | vtoclbg.3 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
5 | 3, 4 | vtoclg 2781 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 |
This theorem is referenced by: pm13.183 2859 sbc8g 2953 sbcco 2967 sbc5 2969 sbcie2g 2979 eqsbc3 2985 sbcng 2986 sbcimg 2987 sbcan 2988 sbcang 2989 sbcor 2990 sbcorg 2991 sbcbig 2992 sbcal 2997 sbcalg 2998 sbcex2 2999 sbcexg 3000 sbcel1v 3008 sbcralg 3024 sbcreug 3026 sbcel12g 3055 sbceqg 3056 csbiebg 3082 elpwg 3561 snssg 3703 preq12bg 3747 elintg 3826 elintrabg 3831 sbcbrg 4030 opelresg 4885 elixpsn 6692 ixpsnf1o 6693 domeng 6709 |
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