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