Metamath Proof Explorer


Theorem nfrex

Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 . See nfrexw for a version with a disjoint variable condition, but not requiring ax-13 . (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 7-Oct-2016) (Proof shortened by Wolf Lammen, 30-Dec-2019) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 nfral.1
 |-  F/_ x A
2 nfral.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 nfrexd
 |-  ( T. -> F/ x E. y e. A ph )
7 6 mptru
 |-  F/ x E. y e. A ph