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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 2140, ax-11 2156, and ax-13 2376. (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 3500 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 2 | vtocl2g.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 3 | 2 | imbi2d 340 | . . 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 3553 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) | 
| 7 | 3, 6 | vtoclg 3553 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒)) | 
| 8 | 1, 7 | mpan9 506 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 | 
| This theorem is referenced by: vtocl3g 3574 vtocl4g 3584 opthg 5481 opelopabsb 5534 vtoclr 5747 funopg 6599 f1osng 6888 fsng 7156 fnpr2g 7231 unexbOLD 7769 op1stg 8027 op2ndg 8028 xpsneng 9097 xpcomeng 9105 sbth 9134 sbthfi 9240 unxpdom 9290 prcdnq 11034 mhmlem 19081 carsgmon 34317 brimageg 35929 brdomaing 35937 brrangeg 35938 rankung 36168 mbfresfi 37674 zindbi 42963 2sbc6g 44439 2sbc5g 44440 fmulcl 45601 | 
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