Metamath Proof Explorer


Theorem mdandyv11

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016)

Ref Expression
Hypotheses mdandyv11.1 ( 𝜑 ↔ ⊥ )
mdandyv11.2 ( 𝜓 ↔ ⊤ )
mdandyv11.3 ( 𝜒 ↔ ⊤ )
mdandyv11.4 ( 𝜃 ↔ ⊤ )
mdandyv11.5 ( 𝜏 ↔ ⊥ )
mdandyv11.6 ( 𝜂 ↔ ⊤ )
Assertion mdandyv11 ( ( ( ( 𝜒𝜓 ) ∧ ( 𝜃𝜓 ) ) ∧ ( 𝜏𝜑 ) ) ∧ ( 𝜂𝜓 ) )

Proof

Step Hyp Ref Expression
1 mdandyv11.1 ( 𝜑 ↔ ⊥ )
2 mdandyv11.2 ( 𝜓 ↔ ⊤ )
3 mdandyv11.3 ( 𝜒 ↔ ⊤ )
4 mdandyv11.4 ( 𝜃 ↔ ⊤ )
5 mdandyv11.5 ( 𝜏 ↔ ⊥ )
6 mdandyv11.6 ( 𝜂 ↔ ⊤ )
7 3 2 bothtbothsame ( 𝜒𝜓 )
8 4 2 bothtbothsame ( 𝜃𝜓 )
9 7 8 pm3.2i ( ( 𝜒𝜓 ) ∧ ( 𝜃𝜓 ) )
10 5 1 bothfbothsame ( 𝜏𝜑 )
11 9 10 pm3.2i ( ( ( 𝜒𝜓 ) ∧ ( 𝜃𝜓 ) ) ∧ ( 𝜏𝜑 ) )
12 6 2 bothtbothsame ( 𝜂𝜓 )
13 11 12 pm3.2i ( ( ( ( 𝜒𝜓 ) ∧ ( 𝜃𝜓 ) ) ∧ ( 𝜏𝜑 ) ) ∧ ( 𝜂𝜓 ) )