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Mirrors > Home > MPE Home > Th. List > rspc2dv | Structured version Visualization version GIF version |
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 6-Mar-2025.) |
Ref | Expression |
---|---|
rspc2dv.1 | ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) |
rspc2dv.2 | ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒)) |
rspc2dv.3 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) |
rspc2dv.4 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
rspc2dv.5 | ⊢ (𝜑 → 𝐵 ∈ 𝐷) |
Ref | Expression |
---|---|
rspc2dv | ⊢ (𝜑 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspc2dv.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
2 | rspc2dv.5 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷) | |
3 | rspc2dv.3 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) | |
4 | rspc2dv.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃)) | |
5 | rspc2dv.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒)) | |
6 | 4, 5 | rspc2va 3623 | . 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) → 𝜒) |
7 | 1, 2, 3, 6 | syl21anc 835 | 1 ⊢ (𝜑 → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 |
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 df-ral 3061 |
This theorem is referenced by: mulscom 27953 addsdilem3 27967 addsdilem4 27968 mulsasslem3 27979 cvxsconn 34699 |
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