Theorem ifex 4630

Description: Existence of the conditional operator (inference form). (Contributed by NM, 2-Sep-2004.)

Hypotheses

Ref	Expression
ifex.1	|- A e. _V
ifex.2	|- B e. _V

Assertion

Ref	Expression
ifex	|- if ( ph , A , B ) e. _V

Proof of Theorem ifex

Step	Hyp	Ref	Expression
1		ifex.1	. 2 |- A e. _V
2		ifex.2	. 2 |- B e. _V
3		ifexg 4629	. 2 |- ( ( A e. _V /\ B e. _V ) -> if ( ph , A , B ) e. _V )
4	1, 2, 3	mp2an 430	1 |- if ( ph , A , B ) e. _V

Syntax hints:   e. wcel 2209   _Vcvv 2821   ifcif 3638

This theorem was proved from 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-14 2212  ax-ext 2220  ax-sep 4247  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-uni 3934

This theorem is referenced by:  elply2  15819  pw1map  17008  nnnninfex  17039