Theorem sbcbidv 3101

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypothesis

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

Assertion

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

Distinct variable group:   ph,x

Allowed substitution hints:   ps(x)   ch(x)   A(x)

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 |- F/xph
2		sbcbidv.1	. 2 |- (ph -> (ps <-> ch))
3	1, 2	sbcbid 3100	1 |- (ph -> ( [.A / x ].ps <-> [.A / x ].ch))

Syntax hints:   -> wi 4   <-> wb 176   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 3048

This theorem is referenced by:  sbcbii  3102  csbcomg  3160  opelopabsb  4698  opelopabf  4712