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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 2750 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) |
3 | abbi 2809 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 = wceq 1542 {cab 2714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 |
This theorem is referenced by: abpr 41755 abtp 41756 |
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