Metamath Proof Explorer

Theorem mdandyv11

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 mdandyv11.1 ⊢φ↔⊥
mdandyv11.2 ⊢ψ↔⊤
mdandyv11.3 ⊢χ↔⊤
mdandyv11.4 ⊢θ↔⊤
mdandyv11.5 ⊢τ↔⊥
mdandyv11.6 ⊢η↔⊤
Assertion mdandyv11 ⊢χ↔ψ∧θ↔ψ∧τ↔φ∧η↔ψ

Proof

Step Hyp Ref Expression
1 mdandyv11.1 ⊢φ↔⊥
2 mdandyv11.2 ⊢ψ↔⊤
3 mdandyv11.3 ⊢χ↔⊤
4 mdandyv11.4 ⊢θ↔⊤
5 mdandyv11.5 ⊢τ↔⊥
6 mdandyv11.6 ⊢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 2 bothtbothsame ⊢η↔ψ
13 11 12 pm3.2i ⊢χ↔ψ∧θ↔ψ∧τ↔φ∧η↔ψ