Theorem onsucelsucexmidlem1 4594

Description: Lemma for onsucelsucexmid 4596. (Contributed by Jim Kingdon, 2-Aug-2019.)

Assertion

Ref	Expression
onsucelsucexmidlem1	|- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Distinct variable group:   ph, x

Proof of Theorem onsucelsucexmidlem1

Step	Hyp	Ref	Expression
1		0ex 4187	. . 3 |- (/) e. _V
2	1	prid1 3749	. 2 |- (/) e. { (/) , { (/) } }
3		eqid 2207	. . 3 |- (/) = (/)
4	3	orci 733	. 2 |- ( (/) = (/) \/ ph )
5		eqeq1 2214	. . . 4 |- ( x = (/) -> ( x = (/) <-> (/) = (/) ) )
6	5	orbi1d 793	. . 3 |- ( x = (/) -> ( ( x = (/) \/ ph ) <-> ( (/) = (/) \/ ph ) ) )
7	6	elrab 2936	. 2 |- ( (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) } <-> ( (/) e. { (/) , { (/) } } /\ ( (/) = (/) \/ ph ) ) )
8	2, 4, 7	mpbir2an 945	1 |- (/) e. { x e. { (/) , { (/) } } | ( x = (/) \/ ph ) }

Syntax hints:   \/ wo 710   = wceq 1373   e. wcel 2178   {crab 2490   (/)c0 3468   {csn 3643   {cpr 3644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-nul 4186

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-nul 3469  df-sn 3649  df-pr 3650

This theorem is referenced by:  onsucelsucexmidlem  4595  onsucelsucexmid  4596  acexmidlem2  5964