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Metamath Proof Explorer

Theorem exan

Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993) (Proof shortened by Andrew Salmon, 25-May-2011) (Proof shortened by Wolf Lammen, 13-Jan-2018) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021) (Proof shortened by Wolf Lammen, 6-Nov-2022) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023)

Ref Expression
Hypotheses exan.1 
|- E. x ph
exan.2 
|- ps
Assertion exan 
|- E. x ( ph /\ ps )

Proof

Step Hyp Ref Expression
1 exan.1 
 |-  E. x ph
2 exan.2 
 |-  ps
3 2 jctr 
 |-  ( ph -> ( ph /\ ps ) )
4 1 3 eximii 
 |-  E. x ( ph /\ ps )