Metamath Proof Explorer


Theorem cnvin

Description: Distributive law for converse over intersection. Theorem 15 of Suppes p. 62. (Contributed by NM, 25-Mar-1998) (Revised by Mario Carneiro, 26-Jun-2014)

Ref Expression
Assertion cnvin AB-1=A-1B-1

Proof

Step Hyp Ref Expression
1 cnvdif AAB-1=A-1AB-1
2 cnvdif AB-1=A-1B-1
3 2 difeq2i A-1AB-1=A-1A-1B-1
4 1 3 eqtri AAB-1=A-1A-1B-1
5 dfin4 AB=AAB
6 5 cnveqi AB-1=AAB-1
7 dfin4 A-1B-1=A-1A-1B-1
8 4 6 7 3eqtr4i AB-1=A-1B-1