Theorem sbcbii 3101

Description: Formula-building inference rule for class substitution. (Contributed by NM, 11-Nov-2005.)

Hypothesis

Ref	Expression
sbcbii.1	|- (ph <-> ps)

Assertion

Ref	Expression
sbcbii	|- ( [.A / x ].ph <-> [.A / x ].ps)

Proof of Theorem sbcbii

Step	Hyp	Ref	Expression
1		sbcbii.1	. . . 4 |- (ph <-> ps)
2	1	a1i 10	. . 3 |- ( T. -> (ph <-> ps))
3	2	sbcbidv 3100	. 2 |- ( T. -> ( [.A / x ].ph <-> [.A / x ].ps))
4	3	trud 1323	1 |- ( [.A / x ].ph <-> [.A / x ].ps)

Syntax hints:   <-> wb 176   T. wtru 1316   [.wsbc 3046

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3047

This theorem is referenced by:  sbcbiiOLD  3102  eqsbc3r  3103  sbccomlem  3116  sbccom  3117  sbcrext  3119  sbcabel  3123  csbco  3145  sbcnel12g  3153  sbcne12g  3154  sbccsbg  3164  sbccsb2g  3165  csbnestgf  3184  csbabg  3197  sbcss  3660  inopab  4862  eqerlem  5960