Theorem bj-ax8 32887

Description: Proof of ax-8 1992 from df-clel 2618 (and FOL). This shows that df-clel 2618 is "too powerful". A possible definition is given by bj-df-clel 32888. (Contributed by BJ, 27-Jun-2019.) Also a direct consequence of eleq1w 2684, which has essentially the same proof. (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ax8	|- ( x = y -> ( x e. z -> y e. z ) )

Proof of Theorem bj-ax8

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 1953	. . . . 5 |- ( x = y -> ( u = x <-> u = y ) )
2	1	anbi1d 741	. . . 4 |- ( x = y -> ( ( u = x /\ u e. z ) <-> ( u = y /\ u e. z ) ) )
3	2	exbidv 1850	. . 3 |- ( x = y -> ( E. u ( u = x /\ u e. z ) <-> E. u ( u = y /\ u e. z ) ) )
4		df-clel 2618	. . 3 |- ( x e. z <-> E. u ( u = x /\ u e. z ) )
5		df-clel 2618	. . 3 |- ( y e. z <-> E. u ( u = y /\ u e. z ) )
6	3, 4, 5	3bitr4g 303	. 2 |- ( x = y -> ( x e. z <-> y e. z ) )
7	6	biimpd 219	1 |- ( x = y -> ( x e. z -> y e. z ) )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-clel 2618

This theorem is referenced by: (None)