Theorem nfcjust 2166

Description: Justification theorem for df-nfc 2167. (Contributed by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
nfcjust	|- ( A. y F/ x y e. A <-> A. z F/ x z e. A )

Distinct variable groups:   x, y, z   y, A, z

Allowed substitution hint:   A( x)

Proof of Theorem nfcjust

Step	Hyp	Ref	Expression
1		nfv 1421	. . 3 |- F/ x y = z
2		eleq1 2100	. . 3 |- ( y = z -> ( y e. A <-> z e. A ) )
3	1, 2	nfbidf 1432	. 2 |- ( y = z -> ( F/ x y e. A <-> F/ x z e. A ) )
4	3	cbvalv 1794	1 |- ( A. y F/ x y e. A <-> A. z F/ x z e. A )

Syntax hints:   <-> wb 98   A.wal 1241   F/wnf 1349   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)