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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 515 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ∧ (𝜓 → 𝜒))) |
5 | 1, 4 | alrimih 1831 | . 2 ⊢ (𝜑 → ∀𝑥(𝐴 = 𝐵 ∧ (𝜓 → 𝜒))) |
6 | bj-gabss 34809 | . 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 399 ∀wal 1541 = wceq 1543 ⊆ wss 3853 {bj-cgab 34807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-v 3400 df-in 3860 df-ss 3870 df-bj-gab 34808 |
This theorem is referenced by: bj-gabeqd 34811 |
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