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