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Mirrors > Home > MPE Home > Th. List > vtocl2g | Structured version Visualization version GIF version |
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2144, ax-11 2161, and ax-13 2371. (Revised by Steven Nguyen, 29-Nov-2022.) |
Ref | Expression |
---|---|
vtocl2g.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2g.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2g.3 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl2g | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3416 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl2g.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 344 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒))) |
4 | vtocl2g.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | vtocl2g.3 | . . . 4 ⊢ 𝜑 | |
6 | 4, 5 | vtoclg 3471 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) |
7 | 3, 6 | vtoclg 3471 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒)) |
8 | 1, 7 | mpan9 510 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 |
This theorem is referenced by: vtocl3g 3477 vtocl4g 3485 uniprgOLD 4817 intprgOLD 4873 opthg 5336 opelopabsb 5386 vtoclr 5587 elimasng 5930 funopg 6374 f1osng 6659 fsng 6910 fnpr2g 6984 unexb 7490 op1stg 7727 op2ndg 7728 xpsneng 8652 xpcomeng 8659 sbth 8688 unxpdom 8805 fpwwe2lem4 10135 prcdnq 10494 mhmlem 18338 carsgmon 31851 brimageg 33867 brdomaing 33875 brrangeg 33876 rankung 34106 mbfresfi 35443 zindbi 40332 2sbc6g 41563 2sbc5g 41564 fmulcl 42656 |
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