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Mirrors > Home > MPE Home > Th. List > vtocl2ga | Structured version Visualization version GIF version |
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 20-Aug-1995.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 23-Aug-2023.) |
Ref | Expression |
---|---|
vtocl2ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl2ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl2ga.3 | ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) |
Ref | Expression |
---|---|
vtocl2ga | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtocl2ga.2 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
2 | 1 | imbi2d 340 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒))) |
3 | vtocl2ga.1 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 340 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓))) |
5 | vtocl2ga.3 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) | |
6 | 5 | ex 413 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑)) |
7 | 4, 6 | vtoclga 3565 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓)) |
8 | 7 | com12 32 | . . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓)) |
9 | 2, 8 | vtoclga 3565 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒)) |
10 | 9 | impcom 408 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 |
This theorem is referenced by: solin 5613 caovcan 7613 xpord2pred 8133 pwfseqlem2 10656 mulcanenq 10957 ltaddnq 10971 ltrnq 10976 genpv 10996 wrdind 14674 fsumrelem 15755 imasleval 17489 fullfunc 17859 fthfunc 17860 symggrplem 18767 pf1ind 21881 mretopd 22603 dvlip 25517 scvxcvx 26497 issubgoilem 30551 cnre2csqlem 32959 |
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