Metamath Proof Explorer

Theorem mdandyv14

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ph, ps accordingly. (Contributed by Jarvin Udandy, 6-Sep-2016)

Ref Expression
Hypotheses mdandyv14.1 ⊢φ↔⊥
mdandyv14.2 ⊢ψ↔⊤
mdandyv14.3 ⊢χ↔⊥
mdandyv14.4 ⊢θ↔⊤
mdandyv14.5 ⊢τ↔⊤
mdandyv14.6 ⊢η↔⊤
Assertion mdandyv14 ⊢χ↔φ∧θ↔ψ∧τ↔ψ∧η↔ψ

Proof

Step Hyp Ref Expression
1 mdandyv14.1 ⊢φ↔⊥
2 mdandyv14.2 ⊢ψ↔⊤
3 mdandyv14.3 ⊢χ↔⊥
4 mdandyv14.4 ⊢θ↔⊤
5 mdandyv14.5 ⊢τ↔⊤
6 mdandyv14.6 ⊢η↔⊤
7 3 1 bothfbothsame ⊢χ↔φ
8 4 2 bothtbothsame ⊢θ↔ψ
9 7 8 pm3.2i ⊢χ↔φ∧θ↔ψ
10 5 2 bothtbothsame ⊢τ↔ψ
11 9 10 pm3.2i ⊢χ↔φ∧θ↔ψ∧τ↔ψ
12 6 2 bothtbothsame ⊢η↔ψ
13 11 12 pm3.2i ⊢χ↔φ∧θ↔ψ∧τ↔ψ∧η↔ψ