Metamath Proof Explorer

Theorem mdandyvr4

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 mdandyvr4.1 ⊢ φ ↔ ζ
mdandyvr4.2 ⊢ ψ ↔ σ
mdandyvr4.3 ⊢ χ ↔ φ
mdandyvr4.4 ⊢ θ ↔ φ
mdandyvr4.5 ⊢ τ ↔ ψ
mdandyvr4.6 ⊢ η ↔ φ
Assertion mdandyvr4 ⊢ χ ↔ ζ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ ζ

Proof

Step Hyp Ref Expression
1 mdandyvr4.1 ⊢ φ ↔ ζ
2 mdandyvr4.2 ⊢ ψ ↔ σ
3 mdandyvr4.3 ⊢ χ ↔ φ
4 mdandyvr4.4 ⊢ θ ↔ φ
5 mdandyvr4.5 ⊢ τ ↔ ψ
6 mdandyvr4.6 ⊢ η ↔ φ
7 3 1 bitri ⊢ χ ↔ ζ
8 4 1 bitri ⊢ θ ↔ ζ
9 7 8 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ ζ
10 5 2 bitri ⊢ τ ↔ σ
11 9 10 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ ζ ∧ τ ↔ σ
12 6 1 bitri ⊢ η ↔ ζ
13 11 12 pm3.2i ⊢ χ ↔ ζ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ ζ