Metamath Proof Explorer

Theorem mdandyv2

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 mdandyv2.1 ⊢ φ ↔ ⊥
mdandyv2.2 ⊢ ψ ↔ ⊤
mdandyv2.3 ⊢ χ ↔ ⊥
mdandyv2.4 ⊢ θ ↔ ⊤
mdandyv2.5 ⊢ τ ↔ ⊥
mdandyv2.6 ⊢ η ↔ ⊥
Assertion mdandyv2 ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv2.1 ⊢ φ ↔ ⊥
2 mdandyv2.2 ⊢ ψ ↔ ⊤
3 mdandyv2.3 ⊢ χ ↔ ⊥
4 mdandyv2.4 ⊢ θ ↔ ⊤
5 mdandyv2.5 ⊢ τ ↔ ⊥
6 mdandyv2.6 ⊢ η ↔ ⊥
7 3 1 bothfbothsame ⊢ χ ↔ φ
8 4 2 bothtbothsame ⊢ θ ↔ ψ
9 7 8 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ
10 5 1 bothfbothsame ⊢ τ ↔ φ
11 9 10 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ
12 6 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ ψ ∧ τ ↔ φ ∧ η ↔ φ