Metamath Proof Explorer

Theorem mdandyv4

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 mdandyv4.1 ⊢ φ ↔ ⊥
mdandyv4.2 ⊢ ψ ↔ ⊤
mdandyv4.3 ⊢ χ ↔ ⊥
mdandyv4.4 ⊢ θ ↔ ⊥
mdandyv4.5 ⊢ τ ↔ ⊤
mdandyv4.6 ⊢ η ↔ ⊥
Assertion mdandyv4 ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ ψ ∧ η ↔ φ

Proof

Step Hyp Ref Expression
1 mdandyv4.1 ⊢ φ ↔ ⊥
2 mdandyv4.2 ⊢ ψ ↔ ⊤
3 mdandyv4.3 ⊢ χ ↔ ⊥
4 mdandyv4.4 ⊢ θ ↔ ⊥
5 mdandyv4.5 ⊢ τ ↔ ⊤
6 mdandyv4.6 ⊢ η ↔ ⊥
7 3 1 bothfbothsame ⊢ χ ↔ φ
8 4 1 bothfbothsame ⊢ θ ↔ φ
9 7 8 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ
10 5 2 bothtbothsame ⊢ τ ↔ ψ
11 9 10 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ ψ
12 6 1 bothfbothsame ⊢ η ↔ φ
13 11 12 pm3.2i ⊢ χ ↔ φ ∧ θ ↔ φ ∧ τ ↔ ψ ∧ η ↔ φ