Metamath Proof Explorer

Theorem mdandyv8

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 mdandyv8.1 ⊢ φ ↔ ⊥
mdandyv8.2 ⊢ ψ ↔ ⊤
mdandyv8.3 ⊢ χ ↔ ⊥
mdandyv8.4 ⊢ θ ↔ ⊥
mdandyv8.5 ⊢ τ ↔ ⊥
mdandyv8.6 ⊢ η ↔ ⊤
Assertion mdandyv8 ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ φ ∧ η ↔ ψ

Proof

Step Hyp Ref Expression
1 mdandyv8.1 ⊢ φ ↔ ⊥
2 mdandyv8.2 ⊢ ψ ↔ ⊤
3 mdandyv8.3 ⊢ χ ↔ ⊥
4 mdandyv8.4 ⊢ θ ↔ ⊥
5 mdandyv8.5 ⊢ τ ↔ ⊥
6 mdandyv8.6 ⊢ η ↔ ⊤
7 3 1 bothfbothsame ⊢ χ ↔ φ
8 4 1 bothfbothsame ⊢ θ ↔ φ
9 7 8 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ
10 5 1 bothfbothsame ⊢ τ ↔ φ
11 9 10 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ φ
12 6 2 bothtbothsame ⊢ η ↔ ψ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ φ ∧ η ↔ ψ