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Mirrors > Home > MPE Home > Th. List > eqab | Structured version Visualization version GIF version |
Description: One direction of eqabb 2869 is provable from fewer axioms. (Contributed by Wolf Lammen, 13-Feb-2025.) |
Ref | Expression |
---|---|
eqab | ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid1 2866 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
2 | abbi 2796 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → {𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥 ∣ 𝜑}) | |
3 | 1, 2 | eqtrid 2780 | 1 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑) → 𝐴 = {𝑥 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 = wceq 1534 ∈ wcel 2099 {cab 2705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 |
This theorem is referenced by: (None) |
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