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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 235 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
4 | vtoclbg.3 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
5 | 3, 4 | vtoclg 2812 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 |
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 |
This theorem is referenced by: pm13.183 2890 sbc8g 2985 sbcco 2999 sbc5 3001 sbcie2g 3011 eqsbc1 3017 sbcng 3018 sbcimg 3019 sbcan 3020 sbcang 3021 sbcor 3022 sbcorg 3023 sbcbig 3024 sbcal 3029 sbcalg 3030 sbcex2 3031 sbcexg 3032 sbcel1v 3040 sbcralg 3056 sbcreug 3058 sbcel12g 3087 sbceqg 3088 csbiebg 3114 elpwg 3598 snssgOLD 3743 preq12bg 3788 elintg 3867 elintrabg 3872 sbcbrg 4072 opelresg 4932 elixpsn 6762 ixpsnf1o 6763 domeng 6779 |
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