Metamath Proof Explorer


Theorem brtxpsd2

Description: Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014)

Ref Expression
Hypotheses brtxpsd2.1 A V
brtxpsd2.2 B V
brtxpsd2.3 R = C ran V E S V
brtxpsd2.4 A C B
Assertion brtxpsd2 A R B x x B x S A

Proof

Step Hyp Ref Expression
1 brtxpsd2.1 A V
2 brtxpsd2.2 B V
3 brtxpsd2.3 R = C ran V E S V
4 brtxpsd2.4 A C B
5 3 breqi A R B A C ran V E S V B
6 brdif A C ran V E S V B A C B ¬ A ran V E S V B
7 5 6 bitri A R B A C B ¬ A ran V E S V B
8 4 7 mpbiran A R B ¬ A ran V E S V B
9 1 2 brtxpsd ¬ A ran V E S V B x x B x S A
10 8 9 bitri A R B x x B x S A