Theorem sbcan 3089

Description: Distribution of class substitution over conjunction. (Contributed by NM, 31-Dec-2016.)

Assertion

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

Proof of Theorem sbcan

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3056	. 2 |- ( [.A / x ].(ph /\ ps) -> A e. _V)
2		sbcex 3056	. . 3 |- ( [.A / x ].ps -> A e. _V)
3	2	adantl 452	. 2 |- (( [.A / x ].ph /\ [.A / x ].ps) -> A e. _V)
4		dfsbcq2 3050	. . 3 |- (y = A -> ([y / x](ph /\ ps) <-> [.A / x ].(ph /\ ps)))
5		dfsbcq2 3050	. . . 4 |- (y = A -> ([y / x]ph <-> [.A / x ].ph))
6		dfsbcq2 3050	. . . 4 |- (y = A -> ([y / x]ps <-> [.A / x ].ps))
7	5, 6	anbi12d 691	. . 3 |- (y = A -> (([y / x]ph /\ [y / x]ps) <-> ( [.A / x ].ph /\ [.A / x ].ps)))
8		sban 2069	. . 3 |- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
9	4, 7, 8	vtoclbg 2916	. 2 |- (A e. _V -> ( [.A / x ].(ph /\ ps) <-> ( [.A / x ].ph /\ [.A / x ].ps)))
10	1, 3, 9	pm5.21nii 342	1 |- ( [.A / x ].(ph /\ ps) <-> ( [.A / x ].ph /\ [.A / x ].ps))

Syntax hints:   <-> wb 176   /\ wa 358   = wceq 1642  [wsb 1648   e. wcel 1710  _Vcvv 2860   [.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:  inopab  4863