Theorem bj-elequ12 32668

Description: An identity law for the non-logical predicate, which combines elequ1 1997 and elequ2 2004. For the analogous theorems for class terms, see eleq1 2689, eleq2 2690 and eleq12 2691. (TODO: move to main part.) (Contributed by BJ, 29-Sep-2019.)

Assertion

Ref	Expression
bj-elequ12	|- ( ( x = y /\ z = t ) -> ( x e. z <-> y e. t ) )

Proof of Theorem bj-elequ12

Step	Hyp	Ref	Expression
1		elequ1 1997	. 2 |- ( x = y -> ( x e. z <-> y e. z ) )
2		elequ2 2004	. 2 |- ( z = t -> ( y e. z <-> y e. t ) )
3	1, 2	sylan9bb 736	1 |- ( ( x = y /\ z = t ) -> ( x e. z <-> y e. t ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

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  ax-8 1992  ax-9 1999

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

This theorem is referenced by:  bj-ru0  32932