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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-gabss | Structured version Visualization version GIF version |
Description: Inclusion of generalized class abstractions. (Contributed by BJ, 4-Oct-2024.) |
Ref | Expression |
---|---|
bj-gabss | ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2730 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝑦 ↔ 𝐵 = 𝑦)) | |
2 | 1 | biimpd 228 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝑦 → 𝐵 = 𝑦)) |
3 | 2 | adantr 479 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → (𝐴 = 𝑦 → 𝐵 = 𝑦)) |
4 | simpr 483 | . . . . . 6 ⊢ ((𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → (𝜑 → 𝜓)) | |
5 | 3, 4 | anim12d 607 | . . . . 5 ⊢ ((𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → ((𝐴 = 𝑦 ∧ 𝜑) → (𝐵 = 𝑦 ∧ 𝜓))) |
6 | 5 | aleximi 1827 | . . . 4 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → (∃𝑥(𝐴 = 𝑦 ∧ 𝜑) → ∃𝑥(𝐵 = 𝑦 ∧ 𝜓))) |
7 | 6 | alrimiv 1923 | . . 3 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → ∀𝑦(∃𝑥(𝐴 = 𝑦 ∧ 𝜑) → ∃𝑥(𝐵 = 𝑦 ∧ 𝜓))) |
8 | ss2ab 4056 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} ⊆ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)} ↔ ∀𝑦(∃𝑥(𝐴 = 𝑦 ∧ 𝜑) → ∃𝑥(𝐵 = 𝑦 ∧ 𝜓))) | |
9 | 7, 8 | sylibr 233 | . 2 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} ⊆ {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)}) |
10 | df-bj-gab 36640 | . 2 ⊢ {𝐴 ∣ 𝑥 ∣ 𝜑} = {𝑦 ∣ ∃𝑥(𝐴 = 𝑦 ∧ 𝜑)} | |
11 | df-bj-gab 36640 | . 2 ⊢ {𝐵 ∣ 𝑥 ∣ 𝜓} = {𝑦 ∣ ∃𝑥(𝐵 = 𝑦 ∧ 𝜓)} | |
12 | 9, 10, 11 | 3sstr4g 4025 | 1 ⊢ (∀𝑥(𝐴 = 𝐵 ∧ (𝜑 → 𝜓)) → {𝐴 ∣ 𝑥 ∣ 𝜑} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1532 = wceq 1534 ∃wex 1774 {cab 2703 ⊆ wss 3947 {bj-cgab 36639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ss 3964 df-bj-gab 36640 |
This theorem is referenced by: bj-gabssd 36642 |
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