Metamath Proof Explorer


Theorem cgr3swap13

Description: Permutation law for three-place congruence. (Contributed by Thierry Arnoux, 3-Oct-2020)

Ref Expression
Hypotheses tgcgrxfr.p 𝑃 = ( Base ‘ 𝐺 )
tgcgrxfr.m = ( dist ‘ 𝐺 )
tgcgrxfr.i 𝐼 = ( Itv ‘ 𝐺 )
tgcgrxfr.r = ( cgrG ‘ 𝐺 )
tgcgrxfr.g ( 𝜑𝐺 ∈ TarskiG )
tgbtwnxfr.a ( 𝜑𝐴𝑃 )
tgbtwnxfr.b ( 𝜑𝐵𝑃 )
tgbtwnxfr.c ( 𝜑𝐶𝑃 )
tgbtwnxfr.d ( 𝜑𝐷𝑃 )
tgbtwnxfr.e ( 𝜑𝐸𝑃 )
tgbtwnxfr.f ( 𝜑𝐹𝑃 )
tgbtwnxfr.2 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
Assertion cgr3swap13 ( 𝜑 → ⟨“ 𝐶 𝐵 𝐴 ”⟩ ⟨“ 𝐹 𝐸 𝐷 ”⟩ )

Proof

Step Hyp Ref Expression
1 tgcgrxfr.p 𝑃 = ( Base ‘ 𝐺 )
2 tgcgrxfr.m = ( dist ‘ 𝐺 )
3 tgcgrxfr.i 𝐼 = ( Itv ‘ 𝐺 )
4 tgcgrxfr.r = ( cgrG ‘ 𝐺 )
5 tgcgrxfr.g ( 𝜑𝐺 ∈ TarskiG )
6 tgbtwnxfr.a ( 𝜑𝐴𝑃 )
7 tgbtwnxfr.b ( 𝜑𝐵𝑃 )
8 tgbtwnxfr.c ( 𝜑𝐶𝑃 )
9 tgbtwnxfr.d ( 𝜑𝐷𝑃 )
10 tgbtwnxfr.e ( 𝜑𝐸𝑃 )
11 tgbtwnxfr.f ( 𝜑𝐹𝑃 )
12 tgbtwnxfr.2 ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
13 1 2 3 4 5 6 7 8 9 10 11 12 cgr3swap12 ( 𝜑 → ⟨“ 𝐵 𝐴 𝐶 ”⟩ ⟨“ 𝐸 𝐷 𝐹 ”⟩ )
14 1 2 3 4 5 7 6 8 10 9 11 13 cgr3swap23 ( 𝜑 → ⟨“ 𝐵 𝐶 𝐴 ”⟩ ⟨“ 𝐸 𝐹 𝐷 ”⟩ )
15 1 2 3 4 5 7 8 6 10 11 9 14 cgr3swap12 ( 𝜑 → ⟨“ 𝐶 𝐵 𝐴 ”⟩ ⟨“ 𝐹 𝐸 𝐷 ”⟩ )