Metamath Proof Explorer

Theorem mdandyv3

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 mdandyv3.1 ⊢ φ ↔ ⊥
mdandyv3.2 ⊢ ψ ↔ ⊤
mdandyv3.3 ⊢ χ ↔ ⊤
mdandyv3.4 ⊢ θ ↔ ⊤
mdandyv3.5 ⊢ τ ↔ ⊥
mdandyv3.6 ⊢ η ↔ ⊥
Assertion mdandyv3 ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv3.1 ⊢ φ ↔ ⊥
2 mdandyv3.2 ⊢ ψ ↔ ⊤
3 mdandyv3.3 ⊢ χ ↔ ⊤
4 mdandyv3.4 ⊢ θ ↔ ⊤
5 mdandyv3.5 ⊢ τ ↔ ⊥
6 mdandyv3.6 ⊢ η ↔ ⊥
7 3 2 bothtbothsame ⊢ χ ↔ ψ
8 4 2 bothtbothsame ⊢ θ ↔ ψ
9 7 8 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ
10 5 1 bothfbothsame ⊢ τ ↔ φ
11 9 10 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ φ
12 6 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ φ