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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 4686 | . 2 ⊢ {𝑋, 𝑌, 𝑍} = {𝑥 ∣ (𝑥 = 𝑋 ∨ 𝑥 = 𝑌 ∨ 𝑥 = 𝑍)} | |
2 | 1 | abeqabi 42673 | 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 1083 ∀wal 1531 = wceq 1533 {cab 2701 {ctp 4625 |
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-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-un 3946 df-sn 4622 df-pr 4624 df-tp 4626 |
This theorem is referenced by: (None) |
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