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Mirrors > Home > MPE Home > Th. List > Mathboxes > abtp | Structured version Visualization version GIF version |
Description: Condition for a class abstraction to be a triple. (Contributed by RP, 25-Aug-2024.) |
Ref | Expression |
---|---|
abtp | ⊢ ({𝑥 ∣ 𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftp2 4692 | . 2 ⊢ {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)} | |
2 | 1 | abeqabi 42144 | 1 ⊢ ({𝑥 ∣ 𝜑} = {𝑋, 𝑌, 𝑍} ↔ ∀𝑥(𝜑 ↔ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ w3o 1086 ∀wal 1539 = wceq 1541 {cab 2709 {ctp 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-un 3952 df-sn 4628 df-pr 4630 df-tp 4632 |
This theorem is referenced by: (None) |
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