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Mirrors > Home > MPE Home > Th. List > abeq2d | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction (deduction form of abeq2 2719). (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
abeq2d.1 | ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) |
Ref | Expression |
---|---|
abeq2d | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abeq2d.1 | . . 3 ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓}) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜓})) |
3 | abid 2598 | . 2 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓) | |
4 | 2, 3 | syl6bb 275 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: abeq2i 2722 fvelimab 6163 ispridlc 33039 ac6s6 33150 dib1dim 35472 mapsnend 38386 |
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