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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 36919 | . 2 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} ⊆ {𝐵 ∣ 𝑥 ∣ 𝜒}) |
6 | 2 | eqcomd 2741 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐴) |
7 | 3 | biimprd 248 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) |
8 | 1, 6, 7 | bj-gabssd 36919 | . 2 ⊢ (𝜑 → {𝐵 ∣ 𝑥 ∣ 𝜒} ⊆ {𝐴 ∣ 𝑥 ∣ 𝜓}) |
9 | 5, 8 | eqssd 4013 | 1 ⊢ (𝜑 → {𝐴 ∣ 𝑥 ∣ 𝜓} = {𝐵 ∣ 𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∀wal 1535 = wceq 1537 {bj-cgab 36916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ss 3980 df-bj-gab 36917 |
This theorem is referenced by: (None) |
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