Theorem xnegeq 12284

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

Step	Hyp	Ref	Expression
1		eqeq1 2802	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = +∞ ↔ 𝐵 = +∞))
2		eqeq1 2802	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 = -∞ ↔ 𝐵 = -∞))
3		negeq 10563	. . . 4 ⊢ (𝐴 = 𝐵 → -𝐴 = -𝐵)
4	2, 3	ifbieq2d 4301	. . 3 ⊢ (𝐴 = 𝐵 → if(𝐴 = -∞, +∞, -𝐴) = if(𝐵 = -∞, +∞, -𝐵))
5	1, 4	ifbieq2d 4301	. 2 ⊢ (𝐴 = 𝐵 → if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴)) = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵)))
6		df-xneg 12190	. 2 ⊢ -𝑒𝐴 = if(𝐴 = +∞, -∞, if(𝐴 = -∞, +∞, -𝐴))
7		df-xneg 12190	. 2 ⊢ -𝑒𝐵 = if(𝐵 = +∞, -∞, if(𝐵 = -∞, +∞, -𝐵))
8	5, 6, 7	3eqtr4g 2857	1 ⊢ (𝐴 = 𝐵 → -𝑒𝐴 = -𝑒𝐵)

Syntax hints:   → wi 4   = wceq 1653  ifcif 4276  +∞cpnf 10359  -∞cmnf 10360  -cneg 10556  -_𝑒cxne 12187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-rex 3094  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-iota 6063  df-fv 6108  df-ov 6880  df-neg 10558  df-xneg 12190

This theorem is referenced by:  xnegcl  12290  xnegneg  12291  xneg11  12292  xltnegi  12293  xnegid  12315  xnegdi  12324  xsubge0  12337  xlesubadd  12339  xmulneg1  12345  xmulneg2  12346  xmulmnf1  12352  xmulm1  12357  xrsdsval  20109  xrsdsreclblem  20111  xblss2ps  22531  xblss2  22532  xrhmeo  23070  xaddeq0  30029  xrsmulgzz  30187  xrge0npcan  30203  carsgclctunlem2  30890  xnegeqd  40396  xnegeqi  40399  supminfxr2  40431  supminfxrrnmpt  40433