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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 2739 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
3 | abbib 2798 | . 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 205 ∀wal 1531 = wceq 1533 {cab 2703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 |
This theorem is referenced by: abpr 42717 abtp 42718 |
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