Metamath Proof Explorer


Theorem frege72

Description: If property A is hereditary in the R -sequence, if x has property A , and if y is a result of an application of the procedure R to x , then y has property A . Proposition 72 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege72.x X U
frege72.y Y V
Assertion frege72 R hereditary A X A X R Y Y A

Proof

Step Hyp Ref Expression
1 frege72.x X U
2 frege72.y Y V
3 2 frege58c z X R z z A [˙Y / z]˙ X R z z A
4 sbcim1 [˙Y / z]˙ X R z z A [˙Y / z]˙ X R z [˙Y / z]˙ z A
5 sbcbr2g Y V [˙Y / z]˙ X R z X R Y / z z
6 csbvarg Y V Y / z z = Y
7 6 breq2d Y V X R Y / z z X R Y
8 5 7 bitrd Y V [˙Y / z]˙ X R z X R Y
9 2 8 ax-mp [˙Y / z]˙ X R z X R Y
10 sbcel1v [˙Y / z]˙ z A Y A
11 4 9 10 3imtr3g [˙Y / z]˙ X R z z A X R Y Y A
12 3 11 syl z X R z z A X R Y Y A
13 1 frege71 z X R z z A X R Y Y A R hereditary A X A X R Y Y A
14 12 13 ax-mp R hereditary A X A X R Y Y A