Metamath Proof Explorer

Theorem mdandyvr1

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 mdandyvr1.1 ⊢ φ ↔ ζ
mdandyvr1.2 ⊢ ψ ↔ σ
mdandyvr1.3 ⊢ χ ↔ ψ
mdandyvr1.4 ⊢ θ ↔ φ
mdandyvr1.5 ⊢ τ ↔ φ
mdandyvr1.6 ⊢ η ↔ φ
Assertion mdandyvr1 ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ ζ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr1.1 ⊢ φ ↔ ζ
2 mdandyvr1.2 ⊢ ψ ↔ σ
3 mdandyvr1.3 ⊢ χ ↔ ψ
4 mdandyvr1.4 ⊢ θ ↔ φ
5 mdandyvr1.5 ⊢ τ ↔ φ
6 mdandyvr1.6 ⊢ 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 1 bitri ⊢ τ ↔ ζ
11 9 10 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ ζ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ σ ∧ θ ↔ ζ ∧ τ ↔ ζ ∧ η ↔ ζ