Theorem rabid1o 17034

Description: Converting between propositions and corresponding subsets of a singleton. (Contributed by Jim Kingdon, 31-Jul-2026.)

Assertion

Ref	Expression
rabid1o	|- ( { x e. 1o | ph } = 1o <-> ph )

Distinct variable group:   ph, x

Proof of Theorem rabid1o

Step	Hyp	Ref	Expression
1		0lt1o 6713	. . 3 |- (/) e. 1o
2		elex2 2838	. . 3 |- ( (/) e. 1o -> E. y y e. 1o )
3		r19.3rmv 3618	. . 3 |- ( E. y y e. 1o -> ( ph <-> A. x e. 1o ph ) )
4	1, 2, 3	mp2b 8	. 2 |- ( ph <-> A. x e. 1o ph )
5		rabid2 2729	. 2 |- ( 1o = { x e. 1o | ph } <-> A. x e. 1o ph )
6		eqcom 2240	. 2 |- ( 1o = { x e. 1o | ph } <-> { x e. 1o | ph } = 1o )
7	4, 5, 6	3bitr2ri 209	1 |- ( { x e. 1o | ph } = 1o <-> ph )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   {crab 2532   (/)c0 3520   1oc1o 6680

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-nul 4259

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3715  df-suc 4516  df-1o 6687

This theorem is used by:  wexmiddc  17042  wexmiddifxylem  17045