Theorem ab0w 3550

Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. (Contributed by GG, 3-Oct-2024.)

Hypothesis

Ref	Expression
ab0w.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
ab0w	|- ( { x | ph } = (/) <-> A. y -. ps )

Distinct variable groups:   x, y   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem ab0w

Step	Hyp	Ref	Expression
1		dfnul4 3522	. . 3 |- (/) = { x | F. }
2	1	eqeq2i 2249	. 2 |- ( { x | ph } = (/) <-> { x | ph } = { x | F. } )
3		df-clab 2225	. . . . . 6 |- ( y e. { x | F. } <-> [ y / x ] F. )
4		sbv 1949	. . . . . 6 |- ( [ y / x ] F. <-> F. )
5	3, 4	bitri 184	. . . . 5 |- ( y e. { x | F. } <-> F. )
6	5	bibi2i 227	. . . 4 |- ( ( ps <-> y e. { x | F. } ) <-> ( ps <-> F. ) )
7	6	albii 1523	. . 3 |- ( A. y ( ps <-> y e. { x | F. } ) <-> A. y ( ps <-> F. ) )
8		ab0w.1	. . . 4 |- ( x = y -> ( ph <-> ps ) )
9	8	eqabcbw 2376	. . 3 |- ( { x | ph } = { x | F. } <-> A. y ( ps <-> y e. { x | F. } ) )
10		nbfal 1413	. . . 4 |- ( -. ps <-> ( ps <-> F. ) )
11	10	albii 1523	. . 3 |- ( A. y -. ps <-> A. y ( ps <-> F. ) )
12	7, 9, 11	3bitr4i 212	. 2 |- ( { x | ph } = { x | F. } <-> A. y -. ps )
13	2, 12	bitri 184	1 |- ( { x | ph } = (/) <-> A. y -. ps )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   F. wfal 1407   [wsb 1815   e. wcel 2209   {cab 2224   (/)c0 3520

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 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

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-dif 3222  df-nul 3521

This theorem is referenced by:  fsetdmprc0  6940