Metamath Proof Explorer

Theorem mdandyvr6

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016)

Ref Expression
Hypotheses mdandyvr6.1 ⊢ φ ↔ ζ
mdandyvr6.2 ⊢ ψ ↔ σ
mdandyvr6.3 ⊢ χ ↔ φ
mdandyvr6.4 ⊢ θ ↔ ψ
mdandyvr6.5 ⊢ τ ↔ ψ
mdandyvr6.6 ⊢ η ↔ φ
Assertion mdandyvr6 ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ σ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr6.1 ⊢ φ ↔ ζ
2 mdandyvr6.2 ⊢ ψ ↔ σ
3 mdandyvr6.3 ⊢ χ ↔ φ
4 mdandyvr6.4 ⊢ θ ↔ ψ
5 mdandyvr6.5 ⊢ τ ↔ ψ
6 mdandyvr6.6 ⊢ η ↔ φ
7 3 1 bitri ⊢ χ ↔ ζ
8 4 2 bitri ⊢ θ ↔ σ
9 7 8 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ σ
10 5 2 bitri ⊢ τ ↔ σ
11 9 10 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ σ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ σ ∧ η ↔ ζ