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Metamath Proof Explorer

Theorem nfrexw

Description: Bound-variable hypothesis builder for restricted quantification. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) Add disjoint variable condition to avoid ax-13 . See nfrex for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

Ref Expression
Hypotheses nfralw.1 
|- F/_ x A
nfralw.2 
|- F/ x ph
Assertion nfrexw 
|- F/ x E. y e. A ph

Proof

Step Hyp Ref Expression
1 nfralw.1 
 |-  F/_ x A
2 nfralw.2 
 |-  F/ x ph
3 nftru 
 |-  F/ y T.
4 1 a1i 
 |-  ( T. -> F/_ x A )
5 2 a1i 
 |-  ( T. -> F/ x ph )
6 3 4 5 nfrexdw 
 |-  ( T. -> F/ x E. y e. A ph )
7 6 mptru 
 |-  F/ x E. y e. A ph