Theorem nfrexya 2591

Description: Not-free for restricted existential quantification where y and A are distinct. See nfrexw 2589 for a version with x and y distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)

Hypotheses

Ref	Expression
nfralya.1	|- F/_ x A
nfralya.2	|- F/ x ph

Assertion

Ref	Expression
nfrexya	|- F/ x E. y e. A ph

Distinct variable group:   y, A

Allowed substitution hints:   ph( x, y)   A( x)

Proof of Theorem nfrexya

Step	Hyp	Ref	Expression
1		nftru 1519	. . 3 |- F/ y T.
2		nfralya.1	. . . 4 |- F/_ x A
3	2	a1i 9	. . 3 |- ( T. -> F/_ x A )
4		nfralya.2	. . . 4 |- F/ x ph
5	4	a1i 9	. . 3 |- ( T. -> F/ x ph )
6	1, 3, 5	nfrexdya 2586	. 2 |- ( T. -> F/ x E. y e. A ph )
7	6	mptru 1411	1 |- F/ x E. y e. A ph

This proof depends on syntax axioms:   T. wtru 1403   F/wnf 1513   F/_wnfc 2379   E.wrex 2529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534

This theorem is used by:  nfiunya  4040  nffrec  6667  nfsup  7332  caucvgsrlemgt1  8162  zsupcllemstep  10662  nfsum1  12122  bezout  12788