Theorem bnj206 30799

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj206.1	|- ( ph' <-> [. M / n ]. ph )
bnj206.2	|- ( ps' <-> [. M / n ]. ps )
bnj206.3	|- ( ch' <-> [. M / n ]. ch )
bnj206.4	|- M e. _V

Assertion

Ref	Expression
bnj206	|- ( [. M / n ]. ( ph /\ ps /\ ch ) <-> ( ph' /\ ps' /\ ch' ) )

Proof of Theorem bnj206

Step	Hyp	Ref	Expression
1		sbc3an 3494	. 2 |- ( [. M / n ]. ( ph /\ ps /\ ch ) <-> ( [. M / n ]. ph /\ [. M / n ]. ps /\ [. M / n ]. ch ) )
2		bnj206.1	. . . 4 |- ( ph' <-> [. M / n ]. ph )
3	2	bicomi 214	. . 3 |- ( [. M / n ]. ph <-> ph' )
4		bnj206.2	. . . 4 |- ( ps' <-> [. M / n ]. ps )
5	4	bicomi 214	. . 3 |- ( [. M / n ]. ps <-> ps' )
6		bnj206.3	. . . 4 |- ( ch' <-> [. M / n ]. ch )
7	6	bicomi 214	. . 3 |- ( [. M / n ]. ch <-> ch' )
8	3, 5, 7	3anbi123i 1251	. 2 |- ( ( [. M / n ]. ph /\ [. M / n ]. ps /\ [. M / n ]. ch ) <-> ( ph' /\ ps' /\ ch' ) )
9	1, 8	bitri 264	1 |- ( [. M / n ]. ( ph /\ ps /\ ch ) <-> ( ph' /\ ps' /\ ch' ) )

Syntax hints:   <-> wb 196   /\ w3a 1037   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj124  30941  bnj207  30951