# Metamath Proof Explorer

## Theorem drsb1

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of Megill p. 448 (p. 16 of preprint). Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (New usage is discouraged.)

Ref Expression
Assertion drsb1
`|- ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )`

### Proof

Step Hyp Ref Expression
1 equequ1
` |-  ( x = y -> ( x = z <-> y = z ) )`
2 1 sps
` |-  ( A. x x = y -> ( x = z <-> y = z ) )`
3 2 imbi1d
` |-  ( A. x x = y -> ( ( x = z -> ph ) <-> ( y = z -> ph ) ) )`
4 2 anbi1d
` |-  ( A. x x = y -> ( ( x = z /\ ph ) <-> ( y = z /\ ph ) ) )`
5 4 drex1
` |-  ( A. x x = y -> ( E. x ( x = z /\ ph ) <-> E. y ( y = z /\ ph ) ) )`
6 3 5 anbi12d
` |-  ( A. x x = y -> ( ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) ) )`
7 dfsb1
` |-  ( [ z / x ] ph <-> ( ( x = z -> ph ) /\ E. x ( x = z /\ ph ) ) )`
8 dfsb1
` |-  ( [ z / y ] ph <-> ( ( y = z -> ph ) /\ E. y ( y = z /\ ph ) ) )`
9 6 7 8 3bitr4g
` |-  ( A. x x = y -> ( [ z / x ] ph <-> [ z / y ] ph ) )`