Metamath Proof Explorer

Theorem mdandyvr5

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 mdandyvr5.1 ⊢ φ ↔ ζ
mdandyvr5.2 ⊢ ψ ↔ σ
mdandyvr5.3 ⊢ χ ↔ ψ
mdandyvr5.4 ⊢ θ ↔ φ
mdandyvr5.5 ⊢ τ ↔ ψ
mdandyvr5.6 ⊢ η ↔ φ
Assertion mdandyvr5 ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr5.1 ⊢ φ ↔ ζ
2 mdandyvr5.2 ⊢ ψ ↔ σ
3 mdandyvr5.3 ⊢ χ ↔ ψ
4 mdandyvr5.4 ⊢ θ ↔ φ
5 mdandyvr5.5 ⊢ τ ↔ ψ
6 mdandyvr5.6 ⊢ η ↔ φ
7 3 2 bitri ⊢ χ ↔ σ
8 4 1 bitri ⊢ θ ↔ ζ
9 7 8 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ ζ
10 5 2 bitri ⊢ τ ↔ σ
11 9 10 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ σ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ ζ