xrnegcon1d - Intuitionistic Logic Explorer

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Theorem xrnegcon1d 11227

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 9790	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
3	2	eqeq2d 2182	. . . 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 9812	. . . 4 ⊢ (𝜑 → -_𝑒𝐵 ∈ ℝ^*)
7		xneg11 9791	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^*) → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = -_𝑒𝐵))
8	5, 6, 7	syl2anc 409	. . 3 ⊢ (𝜑 → (-_𝑒𝐴 = -_𝑒-_𝑒𝐵 ↔ 𝐴 = -_𝑒𝐵))
9	4, 8	bitr3d 189	. 2 ⊢ (𝜑 → (-_𝑒𝐴 = 𝐵 ↔ 𝐴 = -_𝑒𝐵))
10		eqcom 2172	. 2 ⊢ (𝐴 = -_𝑒𝐵 ↔ -_𝑒𝐵 = 𝐴)
11	9, 10	bitrdi 195	1 ⊢ (𝜑 → (-_𝑒𝐴 = 𝐵 ↔ -_𝑒𝐵 = 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 ∈ wcel 2141 ℝ^*cxr 7953 -_𝑒cxne 9726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885

This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-sub 8092 df-neg 8093 df-xneg 9729

This theorem is referenced by: xrminmax 11228 xrmineqinf 11232