Theorem soeq12d 37608

Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
weeq12d.l	|- ( ph -> R = S )
weeq12d.r	|- ( ph -> A = B )

Assertion

Ref	Expression
soeq12d	|- ( ph -> ( R Or A <-> S Or B ) )

Proof of Theorem soeq12d

Step	Hyp	Ref	Expression
1		weeq12d.l	. . 3 |- ( ph -> R = S )
2		soeq1 5054	. . 3 |- ( R = S -> ( R Or A <-> S Or A ) )
3	1, 2	syl 17	. 2 |- ( ph -> ( R Or A <-> S Or A ) )
4		weeq12d.r	. . 3 |- ( ph -> A = B )
5		soeq2 5055	. . 3 |- ( A = B -> ( S Or A <-> S Or B ) )
6	4, 5	syl 17	. 2 |- ( ph -> ( S Or A <-> S Or B ) )
7	3, 6	bitrd 268	1 |- ( ph -> ( R Or A <-> S Or B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   Or wor 5034

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-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-in 3581  df-ss 3588  df-br 4654  df-po 5035  df-so 5036

This theorem is referenced by: (None)