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Mirrors > Home > MPE Home > Th. List > vtoclbg | Structured version Visualization version 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 334 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ 𝜃))) |
4 | vtoclbg.3 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
5 | 3, 4 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝜒 ↔ 𝜃)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: alexeqg 3302 pm13.183 3313 sbc8g 3410 sbc2or 3411 sbcco 3425 sbc5 3427 sbcie2g 3436 eqsbc3 3442 sbcng 3443 sbcimg 3444 sbcan 3445 sbcor 3446 sbcbig 3447 sbcal 3452 sbcex2 3453 sbcel1v 3462 sbcreu 3482 csbiebg 3522 sbcel12 3935 sbceqg 3936 elpwg 4116 snssg 4268 preq12bg 4326 elintgOLD 4419 elintrabg 4424 sbcbr123 4636 opelresg 5324 inisegn0 5416 funfvima3 6399 elixpsn 7833 ixpsnf1o 7834 domeng 7855 1sdom 8048 rankcf 9478 eldm3 30905 br1steqg 30919 br2ndeqg 30920 elima4 30924 brsset 31166 brbigcup 31175 elfix2 31181 elfunsg 31193 elsingles 31195 funpartlem 31219 ellines 31429 elhf2g 31453 cover2g 32679 sbcrexgOLD 36367 sbcangOLD 37760 sbcorgOLD 37761 sbcalgOLD 37773 sbcexgOLD 37774 sbcel12gOLD 37775 sbcel1gvOLD 38116 |
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