Metamath Proof Explorer

Theorem mdandyv12

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 mdandyv12.1 ⊢φ↔⊥
mdandyv12.2 ⊢ψ↔⊤
mdandyv12.3 ⊢χ↔⊥
mdandyv12.4 ⊢θ↔⊥
mdandyv12.5 ⊢τ↔⊤
mdandyv12.6 ⊢η↔⊤
Assertion mdandyv12 ⊢χ↔φ∧θ↔φ∧τ↔ψ∧η↔ψ

Proof

Step Hyp Ref Expression
1 mdandyv12.1 ⊢φ↔⊥
2 mdandyv12.2 ⊢ψ↔⊤
3 mdandyv12.3 ⊢χ↔⊥
4 mdandyv12.4 ⊢θ↔⊥
5 mdandyv12.5 ⊢τ↔⊤
6 mdandyv12.6 ⊢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 2 bothtbothsame ⊢η↔ψ
13 11 12 pm3.2i ⊢χ↔φ∧θ↔φ∧τ↔ψ∧η↔ψ