Metamath Proof Explorer

Theorem mdandyv13

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 mdandyv13.1 ⊢ φ ↔ ⊥
mdandyv13.2 ⊢ ψ ↔ ⊤
mdandyv13.3 ⊢ χ ↔ ⊤
mdandyv13.4 ⊢ θ ↔ ⊥
mdandyv13.5 ⊢ τ ↔ ⊤
mdandyv13.6 ⊢ η ↔ ⊤
Assertion mdandyv13 ⊢ χ ↔ ψ ∧ θ ↔ φ ∧ τ ↔ ψ ∧ η ↔ ψ

Proof

Step Hyp Ref Expression
1 mdandyv13.1 ⊢ φ ↔ ⊥
2 mdandyv13.2 ⊢ ψ ↔ ⊤
3 mdandyv13.3 ⊢ χ ↔ ⊤
4 mdandyv13.4 ⊢ θ ↔ ⊥
5 mdandyv13.5 ⊢ τ ↔ ⊤
6 mdandyv13.6 ⊢ η ↔ ⊤
7 3 2 bothtbothsame ⊢ χ ↔ ψ
8 4 1 bothfbothsame ⊢ θ ↔ φ
9 7 8 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ φ
10 5 2 bothtbothsame ⊢ τ ↔ ψ
11 9 10 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ φ ∧ τ ↔ ψ
12 6 2 bothtbothsame ⊢ η ↔ ψ
13 11 12 pm3.2i ⊢ χ ↔ ψ ∧ θ ↔ φ ∧ τ ↔ ψ ∧ η ↔ ψ