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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-gabssd | Structured version Visualization version GIF version | ||
| Description: Inclusion of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.) | 
| Ref | Expression | 
|---|---|
| bj-gabssd.nf | ⊢ (𝜑 → ∀𝑥𝜑) | 
| bj-gabssd.c | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| bj-gabssd.f | ⊢ (𝜑 → (𝜓 → 𝜒)) | 
| Ref | Expression | 
|---|---|
| bj-gabssd | ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | bj-gabssd.nf | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 2 | bj-gabssd.c | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | bj-gabssd.f | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
| 4 | 2, 3 | jca 511 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ (𝜓 → 𝜒))) | 
| 5 | 1, 4 | alrimih 1824 | . 2 ⊢ (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒))) | 
| 6 | bj-gabss 36936 | . 2 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒)) → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) | |
| 7 | 5, 6 | syl 17 | 1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ⊆ wss 3951 {bj-cgab 36934 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ss 3968 df-bj-gab 36935 | 
| This theorem is referenced by: bj-gabeqd 36938 | 
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