Metamath Proof Explorer

Theorem mdandyv6

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 mdandyv6.1 ⊢ φ ↔ ⊥
mdandyv6.2 ⊢ ψ ↔ ⊤
mdandyv6.3 ⊢ χ ↔ ⊥
mdandyv6.4 ⊢ θ ↔ ⊤
mdandyv6.5 ⊢ τ ↔ ⊤
mdandyv6.6 ⊢ η ↔ ⊥
Assertion mdandyv6 ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ ψ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv6.1 ⊢ φ ↔ ⊥
2 mdandyv6.2 ⊢ ψ ↔ ⊤
3 mdandyv6.3 ⊢ χ ↔ ⊥
4 mdandyv6.4 ⊢ θ ↔ ⊤
5 mdandyv6.5 ⊢ τ ↔ ⊤
6 mdandyv6.6 ⊢ 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 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ ψ ∧ η ↔ φ