Metamath Proof Explorer

Theorem mdandyv10

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 mdandyv10.1 ⊢ φ ↔ ⊥
mdandyv10.2 ⊢ ψ ↔ ⊤
mdandyv10.3 ⊢ χ ↔ ⊥
mdandyv10.4 ⊢ θ ↔ ⊤
mdandyv10.5 ⊢ τ ↔ ⊥
mdandyv10.6 ⊢ η ↔ ⊤
Assertion mdandyv10 ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ ψ

Proof

Step Hyp Ref Expression
1 mdandyv10.1 ⊢ φ ↔ ⊥
2 mdandyv10.2 ⊢ ψ ↔ ⊤
3 mdandyv10.3 ⊢ χ ↔ ⊥
4 mdandyv10.4 ⊢ θ ↔ ⊤
5 mdandyv10.5 ⊢ τ ↔ ⊥
6 mdandyv10.6 ⊢ η ↔ ⊤
7 3 1 bothfbothsame ⊢ χ ↔ φ
8 4 2 bothtbothsame ⊢ θ ↔ ψ
9 7 8 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ
10 5 1 bothfbothsame ⊢ τ ↔ φ
11 9 10 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ
12 6 2 bothtbothsame ⊢ η ↔ ψ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ ψ