Theorem eqabcbw 2376

Description: Version of eqabcb 2377 using implicit substitution, which requires fewer axioms. (Contributed by TM, 24-Jan-2026.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem eqabcbw

Step	Hyp	Ref	Expression
1		eqabbw.1	. . 3 |- ( x = y -> ( ph <-> ps ) )
2	1	eqabbw 2375	. 2 |- ( A = { x | ph } <-> A. y ( y e. A <-> ps ) )
3		eqcom 2240	. 2 |- ( { x | ph } = A <-> A = { x | ph } )
4		bicom 140	. . 3 |- ( ( ps <-> y e. A ) <-> ( y e. A <-> ps ) )
5	4	albii 1523	. 2 |- ( A. y ( ps <-> y e. A ) <-> A. y ( y e. A <-> ps ) )
6	2, 3, 5	3bitr4i 212	1 |- ( { x | ph } = A <-> A. y ( ps <-> y e. A ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224

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-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-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234

This theorem is referenced by:  ab0w  3550