Theorem sbc19.21g 3111

Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)

Hypothesis

Ref	Expression
sbcgf.1	|- F/xph

Assertion

Ref	Expression
sbc19.21g	|- (A e. V -> ( [.A / x ].(ph -> ps) <-> (ph -> [.A / x ].ps)))

Proof of Theorem sbc19.21g

Step	Hyp	Ref	Expression
1		sbcimg 3088	. 2 |- (A e. V -> ( [.A / x ].(ph -> ps) <-> ( [.A / x ].ph -> [.A / x ].ps)))
2		sbcgf.1	. . . 4 |- F/xph
3	2	sbcgf 3110	. . 3 |- (A e. V -> ( [.A / x ].ph <-> ph))
4	3	imbi1d 308	. 2 |- (A e. V -> (( [.A / x ].ph -> [.A / x ].ps) <-> (ph -> [.A / x ].ps)))
5	1, 4	bitrd 244	1 |- (A e. V -> ( [.A / x ].(ph -> ps) <-> (ph -> [.A / x ].ps)))

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   e. wcel 1710   [.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-or 359  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-nfc 2479  df-v 2862  df-sbc 3048

This theorem is referenced by: (None)