Metamath Proof Explorer

Theorem mdandyv8

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 mdandyv8.1 ⊢φ↔⊥
mdandyv8.2 ⊢ψ↔⊤
mdandyv8.3 ⊢χ↔⊥
mdandyv8.4 ⊢θ↔⊥
mdandyv8.5 ⊢τ↔⊥
mdandyv8.6 ⊢η↔⊤
Assertion mdandyv8 ⊢χ↔φ∧θ↔φ∧τ↔φ∧η↔ψ

Proof

Step Hyp Ref Expression
1 mdandyv8.1 ⊢φ↔⊥
2 mdandyv8.2 ⊢ψ↔⊤
3 mdandyv8.3 ⊢χ↔⊥
4 mdandyv8.4 ⊢θ↔⊥
5 mdandyv8.5 ⊢τ↔⊥
6 mdandyv8.6 ⊢η↔⊤
7 3 1 bothfbothsame ⊢χ↔φ
8 4 1 bothfbothsame ⊢θ↔φ
9 7 8 pm3.2i ⊢χ↔φ∧θ↔φ
10 5 1 bothfbothsame ⊢τ↔φ
11 9 10 pm3.2i ⊢χ↔φ∧θ↔φ∧τ↔φ
12 6 2 bothtbothsame ⊢η↔ψ
13 11 12 pm3.2i ⊢χ↔φ∧θ↔φ∧τ↔φ∧η↔ψ