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Theorem xnegeq 13246
Description: Equality of two extended numbers with -𝑒 in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xnegeq (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Proof of Theorem xnegeq
StepHypRef Expression
1 eqeq1 2739 . . 3 (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2 eqeq1 2739 . . . 4 (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3 negeq 11498 . . . 4 (𝐴 = 𝐵 → -𝐴 = -𝐵)
42, 3ifbieq2d 4557 . . 3 (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
51, 4ifbieq2d 4557 . 2 (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6 df-xneg 13152 . 2 -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7 df-xneg 13152 . 2 -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
85, 6, 73eqtr4g 2800 1 (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  ifcif 4531  +∞cpnf 11290  -∞cmnf 11291  -cneg 11491  -𝑒cxne 13149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-neg 11493  df-xneg 13152
This theorem is referenced by:  xnegcl  13252  xnegneg  13253  xneg11  13254  xltnegi  13255  xnegid  13277  xnegdi  13287  xsubge0  13300  xlesubadd  13302  xmulneg1  13308  xmulneg2  13309  xmulmnf1  13315  xmulm1  13320  xrsdsval  21446  xrsdsreclblem  21448  xblss2ps  24427  xblss2  24428  xrhmeo  24991  xaddeq0  32764  xrsmulgzz  32994  xrge0npcan  33008  carsgclctunlem2  34301  xnegeqd  45387  xnegeqi  45390  supminfxr2  45419  supminfxrrnmpt  45421  liminflbuz2  45771
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