![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eqabcb | Structured version Visualization version GIF version |
Description: Equality of a class variable and a class abstraction. Commuted form of eqabb 2879. (Contributed by NM, 20-Aug-1993.) |
Ref | Expression |
---|---|
eqabcb | ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqabb 2879 | . 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) | |
2 | eqcom 2742 | . 2 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ 𝐴 = {𝑥 ∣ 𝜑}) | |
3 | bicom 222 | . . 3 ⊢ ((𝜑 ↔ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑)) | |
4 | 3 | albii 1816 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑)) |
5 | 1, 2, 4 | 3bitr4i 303 | 1 ⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∀wal 1535 = wceq 1537 ∈ wcel 2106 {cab 2712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: rabeqcOLD 3693 dm0rn0 5938 dffo3 7122 dffo3f 7126 dfsup2 9482 rankf 9832 fmla0xp 35368 dfon3 35874 dfiota3 35905 onsupmaxb 43228 scottabf 44236 |
Copyright terms: Public domain | W3C validator |