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Mirrors > Home > ILE Home > Th. List > neleq12d | GIF version |
Description: Equality theorem for negated membership. (Contributed by FL, 10-Aug-2016.) |
Ref | Expression |
---|---|
neleq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
neleq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
neleq12d | ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neleq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | neleq1 2343 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐶)) |
4 | neleq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | neleq2 2344 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
7 | 3, 6 | bitrd 186 | 1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∉ wnel 2339 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 df-nel 2340 |
This theorem is referenced by: (None) |
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