sbcne12 - Metamath Proof Explorer

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Theorem sbcne12 4383

Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)

**Assertion**
Ref	Expression
sbcne12	⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)

**Proof of Theorem sbcne12**
Step	Hyp	Ref	Expression
1		nne 2965	. . . . . 6 ⊢ (¬ 𝐵 ≠ 𝐶 ↔ 𝐵 = 𝐶)
2	1	sbcbii 3803	. . . . 5 ⊢ ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶)
3	2	a1i 11	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ [𝐴 / 𝑥]𝐵 = 𝐶))
4		sbcng 3794	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝐵 ≠ 𝐶 ↔ ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶))
5		sbceqg 4380	. . . . 5 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶))
6		nne 2965	. . . . 5 ⊢ (¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
7	5, 6	bitr4di 292	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
8	3, 4, 7	3bitr3d 312	. . 3 ⊢ (𝐴 ∈ V → (¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
9	8	con4bid 320	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
10		sbcex 3757	. . . 4 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 → 𝐴 ∈ V)
11	10	con3i 155	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝐵 ≠ 𝐶)
12		csbprc 4377	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
13		csbprc 4377	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐶 = ∅)
14	12, 13	eqtr4d 2804	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)
15	14, 6	sylibr 237	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)
16	11, 15	2falsed 379	. 2 ⊢ (¬ 𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶))
17	9, 16	pm2.61i 184	1 ⊢ ([𝐴 / 𝑥]𝐵 ≠ 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 ≠ ⦋𝐴 / 𝑥⦌𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ↔ wb 209 = wceq 1570 ∈ wcel 2146 ≠ wne 2961 Vcvv 3458 [wsbc 3747 ⦋csb 3856 ∅c0 4289

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-nul 4290

This theorem is used by: 2nreu 4412 disjdsct 33085 cdlemkid3N 41748 cdlemkid4 41749

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