Theorem ifcli 39329

Description: Membership (closure) of a conditional operator. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
ifcli.1	|- A e. C
ifcli.2	|- B e. C

Assertion

Ref	Expression
ifcli	|- if ( ph , A , B ) e. C

Proof of Theorem ifcli

Step	Hyp	Ref	Expression
1		ifcli.1	. 2 |- A e. C
2		ifcli.2	. 2 |- B e. C
3	1, 2	keepel 4155	1 |- if ( ph , A , B ) e. C

Syntax hints:   e. wcel 1990   ifcif 4086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-if 4087

This theorem is referenced by:  limsup10exlem  40004