Metamath Proof Explorer

Theorem mdandyvr2

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 mdandyvr2.1 ⊢ φ ↔ ζ
mdandyvr2.2 ⊢ ψ ↔ σ
mdandyvr2.3 ⊢ χ ↔ φ
mdandyvr2.4 ⊢ θ ↔ ψ
mdandyvr2.5 ⊢ τ ↔ φ
mdandyvr2.6 ⊢ η ↔ φ
Assertion mdandyvr2 ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ ζ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr2.1 ⊢ φ ↔ ζ
2 mdandyvr2.2 ⊢ ψ ↔ σ
3 mdandyvr2.3 ⊢ χ ↔ φ
4 mdandyvr2.4 ⊢ θ ↔ ψ
5 mdandyvr2.5 ⊢ τ ↔ φ
6 mdandyvr2.6 ⊢ 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 1 bitri ⊢ τ ↔ ζ
11 9 10 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ ζ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ σ ∧ τ ↔ ζ ∧ η ↔ ζ