Theorem meetval2lem 17022

Description: Lemma for meetval2 17023 and meeteu 17024. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu into meetlem?

Hypotheses

Ref	Expression
meetval2.b	|- B = ( Base ` K )
meetval2.l	|- .<_ = ( le ` K )
meetval2.m	|- ./\ = ( meet ` K )
meetval2.k	|- ( ph -> K e. V )
meetval2.x	|- ( ph -> X e. B )
meetval2.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
meetval2lem	|- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )

Distinct variable groups:   x, z, B   x, ./\ , z   x, y, K, z   y, .<_   x, X, y, z   x, Y, y, z

Allowed substitution hints:   ph( x, y, z)   B( y)   .<_ ( x, z)   ./\ ( y)   V( x, y, z)

Proof of Theorem meetval2lem

Step	Hyp	Ref	Expression
1		breq2 4657	. . 3 |- ( y = X -> ( x .<_ y <-> x .<_ X ) )
2		breq2 4657	. . 3 |- ( y = Y -> ( x .<_ y <-> x .<_ Y ) )
3	1, 2	ralprg 4234	. 2 |- ( ( X e. B /\ Y e. B ) -> ( A. y e. { X , Y } x .<_ y <-> ( x .<_ X /\ x .<_ Y ) ) )
4		breq2 4657	. . . . 5 |- ( y = X -> ( z .<_ y <-> z .<_ X ) )
5		breq2 4657	. . . . 5 |- ( y = Y -> ( z .<_ y <-> z .<_ Y ) )
6	4, 5	ralprg 4234	. . . 4 |- ( ( X e. B /\ Y e. B ) -> ( A. y e. { X , Y } z .<_ y <-> ( z .<_ X /\ z .<_ Y ) ) )
7	6	imbi1d 331	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } z .<_ y -> z .<_ x ) <-> ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) )
8	7	ralbidv 2986	. 2 |- ( ( X e. B /\ Y e. B ) -> ( A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) <-> A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) )
9	3, 8	anbi12d 747	1 |- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {cpr 4179   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   meetcmee 16945

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-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  meetval2  17023  meeteu  17024