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Metamath Proof Explorer

Theorem 2moswapv

Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 10-Apr-2004) (Revised by GG, 22-Aug-2023) Factor out common proof lines with moexexvw . (Revised by Wolf Lammen, 2-Oct-2023)

Ref Expression
Assertion 2moswapv 
|- ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )

Proof

Step Hyp Ref Expression
1 nfe1 
 |-  F/ y E. y ph
2 1 nfmov 
 |-  F/ y E* x E. y ph
3 nfe1 
 |-  F/ x E. x ( E. y ph /\ ph )
4 3 nfmov 
 |-  F/ x E* y E. x ( E. y ph /\ ph )
5 1 2 4 moexexlem 
 |-  ( ( E* x E. y ph /\ A. x E* y ph ) -> E* y E. x ( E. y ph /\ ph ) )
6 5 expcom 
 |-  ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ( E. y ph /\ ph ) ) )
7 19.8a 
 |-  ( ph -> E. y ph )
8 7 pm4.71ri 
 |-  ( ph <-> ( E. y ph /\ ph ) )
9 8 exbii 
 |-  ( E. x ph <-> E. x ( E. y ph /\ ph ) )
10 9 mobii 
 |-  ( E* y E. x ph <-> E* y E. x ( E. y ph /\ ph ) )
11 6 10 imbitrrdi 
 |-  ( A. x E* y ph -> ( E* x E. y ph -> E* y E. x ph ) )