Metamath Proof Explorer


Theorem prsrcmpltd

Description: If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023)

Ref Expression
Hypotheses prsrcmpltd.1 φ C A D A ψ
prsrcmpltd.2 φ C A D B A ψ
prsrcmpltd.3 φ C B A D A ψ
prsrcmpltd.4 φ C B A D B A ψ
Assertion prsrcmpltd φ C B D B ψ

Proof

Step Hyp Ref Expression
1 prsrcmpltd.1 φ C A D A ψ
2 prsrcmpltd.2 φ C A D B A ψ
3 prsrcmpltd.3 φ C B A D A ψ
4 prsrcmpltd.4 φ C B A D B A ψ
5 1 expdimp φ C A D A ψ
6 2 expdimp φ C A D B A ψ
7 5 6 srcmpltd φ C A D B ψ
8 7 impancom φ D B C A ψ
9 3 expdimp φ C B A D A ψ
10 4 expdimp φ C B A D B A ψ
11 9 10 srcmpltd φ C B A D B ψ
12 11 impancom φ D B C B A ψ
13 8 12 srcmpltd φ D B C B ψ
14 13 ex φ D B C B ψ
15 14 impcomd φ C B D B ψ