Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2141, ax-11 2157, and ax-13 2386. (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 3512 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl2g.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 343 | . . 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 3567 | . . 3 ⊢ (𝐴 ∈ V → 𝜓) |
7 | 3, 6 | vtoclg 3567 | . 2 ⊢ (𝐵 ∈ 𝑊 → (𝐴 ∈ V → 𝜒)) |
8 | 1, 7 | mpan9 509 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 |
This theorem is referenced by: vtocl4g 3578 uniprg 4855 intprg 4909 opthg 5368 opelopabsb 5416 vtoclr 5614 elimasng 5954 funopg 6388 f1osng 6654 fsng 6898 fnpr2g 6972 unexb 7470 op1stg 7700 op2ndg 7701 xpsneng 8601 xpcomeng 8608 sbth 8636 unxpdom 8724 fpwwe2lem5 10055 prcdnq 10414 mhmlem 18218 carsgmon 31572 brimageg 33388 brdomaing 33396 brrangeg 33397 rankung 33627 mbfresfi 34937 zindbi 39541 2sbc6g 40745 2sbc5g 40746 fmulcl 41860 |
Copyright terms: Public domain | W3C validator |