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Mirrors > Home > MPE Home > Th. List > abeq1i | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.) (Proof shortened by Wolf Lammen, 15-Nov-2019.) |
Ref | Expression |
---|---|
abeq1i.1 | ⊢ {𝑥 ∣ 𝜑} = 𝐴 |
Ref | Expression |
---|---|
abeq1i | ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq1i.1 | . . . 4 ⊢ {𝑥 ∣ 𝜑} = 𝐴 | |
2 | 1 | eqcomi 2827 | . . 3 ⊢ 𝐴 = {𝑥 ∣ 𝜑} |
3 | 2 | abeq2i 2945 | . 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑) |
4 | 3 | bicomi 225 | 1 ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cab 2796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-tru 1531 df-ex 1772 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: (None) |
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