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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-gabeqd | Structured version Visualization version GIF version | ||
| Description: Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024.) |
| Ref | Expression |
|---|---|
| bj-gabeqd.nf | ⊢ (𝜑 → ∀𝑥𝜑) |
| bj-gabeqd.c | ⊢ (𝜑 → 𝐴 = 𝐵) |
| bj-gabeqd.f | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| bj-gabeqd | ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bj-gabeqd.nf | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
| 2 | bj-gabeqd.c | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | bj-gabeqd.f | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 4 | 3 | biimpd 229 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
| 5 | 1, 2, 4 | bj-gabssd 36937 | . 2 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) |
| 6 | 2 | eqcomd 2743 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) |
| 7 | 3 | biimprd 248 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) |
| 8 | 1, 6, 7 | bj-gabssd 36937 | . 2 ⊢ (𝜑 → {𝐵 ∣ 𝑥 ∣ 𝜒} ⊆ {𝐴 ∣ 𝑥 ∣ 𝜓}) |
| 9 | 5, 8 | eqssd 4001 | 1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1538 = wceq 1540 {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: (None) |
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