Metamath Proof Explorer

Theorem mdandyv3

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 mdandyv3.1 ⊢φ↔⊥
mdandyv3.2 ⊢ψ↔⊤
mdandyv3.3 ⊢χ↔⊤
mdandyv3.4 ⊢θ↔⊤
mdandyv3.5 ⊢τ↔⊥
mdandyv3.6 ⊢η↔⊥
Assertion mdandyv3 ⊢χ↔ψ∧θ↔ψ∧τ↔φ∧η↔φ

Proof

Step Hyp Ref Expression
1 mdandyv3.1 ⊢φ↔⊥
2 mdandyv3.2 ⊢ψ↔⊤
3 mdandyv3.3 ⊢χ↔⊤
4 mdandyv3.4 ⊢θ↔⊤
5 mdandyv3.5 ⊢τ↔⊥
6 mdandyv3.6 ⊢η↔⊥
7 3 2 bothtbothsame ⊢χ↔ψ
8 4 2 bothtbothsame ⊢θ↔ψ
9 7 8 pm3.2i ⊢χ↔ψ∧θ↔ψ
10 5 1 bothfbothsame ⊢τ↔φ
11 9 10 pm3.2i ⊢χ↔ψ∧θ↔ψ∧τ↔φ
12 6 1 bothfbothsame ⊢η↔φ
13 11 12 pm3.2i ⊢χ↔ψ∧θ↔ψ∧τ↔φ∧η↔φ