Metamath Proof Explorer

Theorem mdandyvr7

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 mdandyvr7.1 ⊢φ↔ζ
mdandyvr7.2 ⊢ψ↔σ
mdandyvr7.3 ⊢χ↔ψ
mdandyvr7.4 ⊢θ↔ψ
mdandyvr7.5 ⊢τ↔ψ
mdandyvr7.6 ⊢η↔φ
Assertion mdandyvr7 ⊢χ↔σ∧θ↔σ∧τ↔σ∧η↔ζ

Proof

Step Hyp Ref Expression
1 mdandyvr7.1 ⊢φ↔ζ
2 mdandyvr7.2 ⊢ψ↔σ
3 mdandyvr7.3 ⊢χ↔ψ
4 mdandyvr7.4 ⊢θ↔ψ
5 mdandyvr7.5 ⊢τ↔ψ
6 mdandyvr7.6 ⊢η↔φ
7 3 2 bitri ⊢χ↔σ
8 4 2 bitri ⊢θ↔σ
9 7 8 pm3.2i ⊢χ↔σ∧θ↔σ
10 5 2 bitri ⊢τ↔σ
11 9 10 pm3.2i ⊢χ↔σ∧θ↔σ∧τ↔σ
12 6 1 bitri ⊢η↔ζ
13 11 12 pm3.2i ⊢χ↔σ∧θ↔σ∧τ↔σ∧η↔ζ