Metamath Proof Explorer

Theorem mdandyv7

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 mdandyv7.1 ⊢φ↔⊥
mdandyv7.2 ⊢ψ↔⊤
mdandyv7.3 ⊢χ↔⊤
mdandyv7.4 ⊢θ↔⊤
mdandyv7.5 ⊢τ↔⊤
mdandyv7.6 ⊢η↔⊥
Assertion mdandyv7 ⊢χ↔ψ∧θ↔ψ∧τ↔ψ∧η↔φ

Proof

Step Hyp Ref Expression
1 mdandyv7.1 ⊢φ↔⊥
2 mdandyv7.2 ⊢ψ↔⊤
3 mdandyv7.3 ⊢χ↔⊤
4 mdandyv7.4 ⊢θ↔⊤
5 mdandyv7.5 ⊢τ↔⊤
6 mdandyv7.6 ⊢η↔⊥
7 3 2 bothtbothsame ⊢χ↔ψ
8 4 2 bothtbothsame ⊢θ↔ψ
9 7 8 pm3.2i ⊢χ↔ψ∧θ↔ψ
10 5 2 bothtbothsame ⊢τ↔ψ
11 9 10 pm3.2i ⊢χ↔ψ∧θ↔ψ∧τ↔ψ
12 6 1 bothfbothsame ⊢η↔φ
13 11 12 pm3.2i ⊢χ↔ψ∧θ↔ψ∧τ↔ψ∧η↔φ