Metamath Proof Explorer


Theorem exbirVD

Description: Virtual deduction proof of exbir . 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 ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( ch <-> th ) ) ).
2:: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) ->. ( ph /\ ps ) ).
3:: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ,. ( ph /\ ps ) , th ->. th ).
5:1,2,?: e12 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( ch <-> th ) ).
6:3,5,?: e32 |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) , th ->. ch ).
7:6: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) , ( ph /\ ps ) ->. ( th -> ch ) ).
8:7: |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ( ph /\ ps ) -> ( th -> ch ) ) ).
9:8,?: e1a |- (. ( ( ph /\ ps ) -> ( ch <-> th ) ) ->. ( ph -> ( ps -> ( th -> ch ) ) ) ).
qed:9: |- ( ( ( ph /\ ps ) -> ( ch <-> th ) ) -> ( ph -> ( ps -> ( th -> ch ) ) ) )
(Contributed by Alan Sare, 13-Dec-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion exbirVD φψχθφψθχ

Proof

Step Hyp Ref Expression
1 idn3 φψχθ,φψ,θθ
2 idn1 φψχθφψχθ
3 idn2 φψχθ,φψφψ
4 id φψχθφψχθ
5 2 3 4 e12 φψχθ,φψχθ
6 biimpr χθθχ
7 6 com12 θχθχ
8 1 5 7 e32 φψχθ,φψ,θχ
9 8 in3 φψχθ,φψθχ
10 9 in2 φψχθφψθχ
11 pm3.3 φψθχφψθχ
12 10 11 e1a φψχθφψθχ
13 12 in1 φψχθφψθχ