Metamath Proof Explorer

Theorem bitr3VD

Description: Virtual deduction proof of bitr3 . The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1:: |- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2:1,?: e1a |- (. ( ph <-> ps ) ->. ( ps <-> ph ) ).
3:: |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ph <-> ch ) ).
4:3,?: e2 |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ch <-> ph ) ).
5:2,4,?: e12 |- (. ( ph <-> ps ) ,. ( ph <-> ch ) ->. ( ps <-> ch ) ).
6:5: |- (. ( ph <-> ps ) ->. ( ( ph <-> ch ) -> ( ps <-> ch ) ) ).
qed:6: |- ( ( ph <-> ps ) -> ( ( ph <-> ch ) -> ( ps <-> ch ) ) )
(Contributed by Alan Sare, 31-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion bitr3VD ⊢ φ ↔ ψ → φ ↔ χ → ψ ↔ χ

Proof

Step Hyp Ref Expression
1 id ⊢ φ ↔ ψ → φ ↔ ψ
2 1 bicomd ⊢ φ ↔ ψ → ψ ↔ φ
3 id ⊢ φ ↔ χ → φ ↔ χ
4 3 bicomd ⊢ φ ↔ χ → χ ↔ φ
5 biantr ⊢ ψ ↔ φ ∧ χ ↔ φ → ψ ↔ χ
6 5 ex ⊢ ψ ↔ φ → χ ↔ φ → ψ ↔ χ
7 2 4 6 syl2im ⊢ φ ↔ ψ → φ ↔ χ → ψ ↔ χ