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| Mirrors > Home > MPE Home > Th. List > eqabcb | Structured version Visualization version GIF version | ||
| Description: Equality of a class variable and a class abstraction. Commuted form of eqabb 2881. (Contributed by NM, 20-Aug-1993.) |
| Ref | Expression |
|---|---|
| eqabcb | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqabb 2881 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
| 2 | eqcom 2744 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ 𝐴 = {𝑥 ∣ 𝜑}) | |
| 3 | bicom 222 | . . 3 ⊢ ((𝜑 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) | |
| 4 | 3 | albii 1819 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
| 5 | 1, 2, 4 | 3bitr4i 303 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {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-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-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 |
| This theorem is referenced by: rabeqcOLD 3690 dm0rn0 5935 dffo3 7122 dffo3f 7126 dfsup2 9484 rankf 9834 fmla0xp 35388 dfon3 35893 dfiota3 35924 onsupmaxb 43251 scottabf 44259 |
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