Description: Virtual deduction proof of csbeq2 . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbeq2 is csbeq2gVD without virtual deductions and was automatically derived from csbeq2gVD .
1:: | |- (. A e. V ->. A e. V ). |
2:1: | |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ). |
3:1: | |- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ). |
4:2,3: | |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ). |
qed:4: | |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) |
Ref | Expression | ||
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Assertion | csbeq2gVD | |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) |
Step | Hyp | Ref | Expression |
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1 | idn1 | |- (. A e. V ->. A e. V ). |
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2 | spsbc | |- ( A e. V -> ( A. x B = C -> [. A / x ]. B = C ) ) |
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3 | 1 2 | e1a | |- (. A e. V ->. ( A. x B = C -> [. A / x ]. B = C ) ). |
4 | sbceqg | |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ) |
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5 | 1 4 | e1a | |- (. A e. V ->. ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) ). |
6 | imbi2 | |- ( ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) -> ( ( A. x B = C -> [. A / x ]. B = C ) <-> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) ) |
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7 | 6 | biimpcd | |- ( ( A. x B = C -> [. A / x ]. B = C ) -> ( ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) ) |
8 | 3 5 7 | e11 | |- (. A e. V ->. ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ). |
9 | 8 | in1 | |- ( A e. V -> ( A. x B = C -> [_ A / x ]_ B = [_ A / x ]_ C ) ) |