Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2426 | . . 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 2427 | . . 3 ⊢ (𝐶 = 𝐷 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝜑 → (𝐵 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
7 | 3, 6 | bitrd 187 | 1 ⊢ (𝜑 → (𝐴 ∉ 𝐶 ↔ 𝐵 ∉ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1335 ∉ wnel 2422 |
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 604 ax-in2 605 ax-5 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-cleq 2150 df-clel 2153 df-nel 2423 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |