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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 3634 | . 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) → 𝜒) | 
| 7 | 1, 2, 3, 6 | syl21anc 838 | 1 ⊢ (𝜑 → 𝜒) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 | 
| This theorem is referenced by: mulscom 28165 addsdilem3 28179 addsdilem4 28180 mulsasslem3 28191 rprmdvds 33547 cvxsconn 35248 oppcmndclem 48905 termcbasmo 49128 fulltermc2 49144 | 
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