Metamath Proof Explorer

Theorem mdandyv7

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 mdandyv7.1 ⊢ φ ↔ ⊥
mdandyv7.2 ⊢ ψ ↔ ⊤
mdandyv7.3 ⊢ χ ↔ ⊤
mdandyv7.4 ⊢ θ ↔ ⊤
mdandyv7.5 ⊢ τ ↔ ⊤
mdandyv7.6 ⊢ η ↔ ⊥
Assertion mdandyv7 ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ ψ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv7.1 ⊢ φ ↔ ⊥
2 mdandyv7.2 ⊢ ψ ↔ ⊤
3 mdandyv7.3 ⊢ χ ↔ ⊤
4 mdandyv7.4 ⊢ θ ↔ ⊤
5 mdandyv7.5 ⊢ τ ↔ ⊤
6 mdandyv7.6 ⊢ η ↔ ⊥
7 3 2 bothtbothsame ⊢ χ ↔ ψ
8 4 2 bothtbothsame ⊢ θ ↔ ψ
9 7 8 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ
10 5 2 bothtbothsame ⊢ τ ↔ ψ
11 9 10 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ ψ
12 6 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ ψ ∧ τ ↔ ψ ∧ η ↔ φ