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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 2136 and ax-11 2153. (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 339 | . . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 → 𝜓) ↔ (𝐴 ∈ 𝐶 → 𝜒))) |
3 | vtocl2ga.1 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 339 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ 𝐷 → 𝜑) ↔ (𝑦 ∈ 𝐷 → 𝜓))) |
5 | vtocl2ga.3 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → 𝜑) | |
6 | 5 | ex 412 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜑)) |
7 | 4, 6 | vtoclga 3566 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝑦 ∈ 𝐷 → 𝜓)) |
8 | 7 | com12 32 | . . 3 ⊢ (𝑦 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜓)) |
9 | 2, 8 | vtoclga 3566 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → 𝜒)) |
10 | 9 | impcom 407 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 |
This theorem is referenced by: solin 5614 caovcan 7614 xpord2pred 8134 pwfseqlem2 10657 mulcanenq 10958 ltaddnq 10972 ltrnq 10977 genpv 10997 wrdind 14677 fsumrelem 15758 imasleval 17492 fullfunc 17862 fthfunc 17863 symggrplem 18802 pf1ind 22095 mretopd 22817 dvlip 25743 scvxcvx 26723 issubgoilem 30777 cnre2csqlem 33185 |
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