xrnegcon1d - Intuitionistic Logic Explorer

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Theorem xrnegcon1d 12030

Description: Contraposition law for extended real unary minus. (Contributed by Jim Kingdon, 2-May-2023.)

**Hypotheses**
Ref	Expression
xrnegcon1d.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrnegcon1d.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)

**Assertion**
Ref	Expression
xrnegcon1d	⊢ (𝜑 → (-_𝑒𝐴 = 𝐵 ↔ -_𝑒𝐵 = 𝐴))

**Proof of Theorem xrnegcon1d**
Step	Hyp	Ref	Expression
1		xrnegcon1d.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
2		xnegneg 10235	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
3	2	eqeq2d 2250	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ -_𝑒𝐴 = 𝐵))
4	1, 3	syl 14	. . 3 ⊢ (𝜑 → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ -_𝑒𝐴 = 𝐵))
5		xrnegcon1d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
6	1	xnegcld 10257	. . . 4 ⊢ (𝜑 → -_𝑒𝐵 ∈ ℝ^*)
7		xneg11 10236	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = -_𝑒𝐵))
8	5, 6, 7	syl2anc 415	. . 3 ⊢ (𝜑 → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = -_𝑒𝐵))
9	4, 8	bitr3d 190	. 2 ⊢ (𝜑 → (-_𝑒𝐴 = 𝐵 ↔ 𝐴 = -_𝑒𝐵))
10		eqcom 2240	. 2 ⊢ (𝐴 = -_𝑒𝐵 ↔ -_𝑒𝐵 = 𝐴)
11	9, 10	bitrdi 196	1 ⊢ (𝜑 → (-_𝑒𝐴 = 𝐵 ↔ -_𝑒𝐵 = 𝐴))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ↔ wb 105 = wceq 1402 ∈ wcel 2209 ℝ^*cxr 8359 -_𝑒cxne 10171

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290

This proof depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-pnf 8362 df-mnf 8363 df-xr 8364 df-sub 8499 df-neg 8500 df-xneg 10174

This theorem is used by: xrminmax 12031 xrmineqinf 12035