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| Mirrors > Home > MPE Home > Th. List > Mathboxes > abeqabi | Structured version Visualization version GIF version | ||
| Description: Generalized condition for a class abstraction to be equal to some class. (Contributed by RP, 2-Sep-2024.) |
| Ref | Expression |
|---|---|
| abeqabi.a | ⊢ 𝐴 = {𝑥 ∣ 𝜓} |
| Ref | Expression |
|---|---|
| abeqabi | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abeqabi.a | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜓} | |
| 2 | 1 | eqeq2i 2750 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
| 3 | abbib 2811 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝜑 ↔ 𝜓)) | |
| 4 | 2, 3 | bitri 275 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 {cab 2714 |
| 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-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-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 |
| This theorem is referenced by: abpr 43422 abtp 43423 |
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