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Mirrors > Home > MPE Home > Th. List > rspcdf | Structured version Visualization version GIF version |
Description: Restricted specialization, using implicit substitution. (Contributed by Emmett Weisz, 16-Jan-2020.) |
Ref | Expression |
---|---|
rspcdf.1 | ⊢ Ⅎ𝑥𝜑 |
rspcdf.2 | ⊢ Ⅎ𝑥𝜒 |
rspcdf.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
rspcdf.4 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rspcdf | ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | rspcdf.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | ex 414 | . . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
4 | 1, 3 | alrimi 2207 | . 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) |
5 | rspcdf.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
6 | rspcdf.2 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
7 | 6 | rspct 3569 | . 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))) |
8 | 4, 5, 7 | sylc 65 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-v 3449 |
This theorem is referenced by: rspc2daf 31446 |
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