Metamath Proof Explorer

Theorem mdandyv0

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 mdandyv0.1 ⊢ φ ↔ ⊥
mdandyv0.2 ⊢ ψ ↔ ⊤
mdandyv0.3 ⊢ χ ↔ ⊥
mdandyv0.4 ⊢ θ ↔ ⊥
mdandyv0.5 ⊢ τ ↔ ⊥
mdandyv0.6 ⊢ η ↔ ⊥
Assertion mdandyv0 ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ φ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv0.1 ⊢ φ ↔ ⊥
2 mdandyv0.2 ⊢ ψ ↔ ⊤
3 mdandyv0.3 ⊢ χ ↔ ⊥
4 mdandyv0.4 ⊢ θ ↔ ⊥
5 mdandyv0.5 ⊢ τ ↔ ⊥
6 mdandyv0.6 ⊢ 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 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ φ ∧ η ↔ φ