Metamath Proof Explorer

Theorem mdandyvr3

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 mdandyvr3.1 ⊢ φ ↔ ζ
mdandyvr3.2 ⊢ ψ ↔ σ
mdandyvr3.3 ⊢ χ ↔ ψ
mdandyvr3.4 ⊢ θ ↔ ψ
mdandyvr3.5 ⊢ τ ↔ φ
mdandyvr3.6 ⊢ η ↔ φ
Assertion mdandyvr3 ⊢ χ ↔ σ ∧ θ ↔ σ ∧ τ ↔ ζ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr3.1 ⊢ φ ↔ ζ
2 mdandyvr3.2 ⊢ ψ ↔ σ
3 mdandyvr3.3 ⊢ χ ↔ ψ
4 mdandyvr3.4 ⊢ θ ↔ ψ
5 mdandyvr3.5 ⊢ τ ↔ φ
6 mdandyvr3.6 ⊢ η ↔ φ
7 3 2 bitri ⊢ χ ↔ σ
8 4 2 bitri ⊢ θ ↔ σ
9 7 8 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ σ
10 5 1 bitri ⊢ τ ↔ ζ
11 9 10 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ σ ∧ τ ↔ ζ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ σ ∧ τ ↔ ζ ∧ η ↔ ζ