New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > pm2.61da3ne | GIF version |
Description: Deduction eliminating three inequalities in an antecedent. (Contributed by NM, 15-Jun-2013.) |
Ref | Expression |
---|---|
pm2.61da3ne.1 | ⊢ ((φ ∧ A = B) → ψ) |
pm2.61da3ne.2 | ⊢ ((φ ∧ C = D) → ψ) |
pm2.61da3ne.3 | ⊢ ((φ ∧ E = F) → ψ) |
pm2.61da3ne.4 | ⊢ ((φ ∧ (A ≠ B ∧ C ≠ D ∧ E ≠ F)) → ψ) |
Ref | Expression |
---|---|
pm2.61da3ne | ⊢ (φ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.61da3ne.1 | . 2 ⊢ ((φ ∧ A = B) → ψ) | |
2 | pm2.61da3ne.2 | . 2 ⊢ ((φ ∧ C = D) → ψ) | |
3 | pm2.61da3ne.3 | . . . 4 ⊢ ((φ ∧ E = F) → ψ) | |
4 | 3 | adantlr 695 | . . 3 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E = F) → ψ) |
5 | simpll 730 | . . . 4 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E ≠ F) → φ) | |
6 | simplrl 736 | . . . 4 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E ≠ F) → A ≠ B) | |
7 | simplrr 737 | . . . 4 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E ≠ F) → C ≠ D) | |
8 | simpr 447 | . . . 4 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E ≠ F) → E ≠ F) | |
9 | pm2.61da3ne.4 | . . . 4 ⊢ ((φ ∧ (A ≠ B ∧ C ≠ D ∧ E ≠ F)) → ψ) | |
10 | 5, 6, 7, 8, 9 | syl13anc 1184 | . . 3 ⊢ (((φ ∧ (A ≠ B ∧ C ≠ D)) ∧ E ≠ F) → ψ) |
11 | 4, 10 | pm2.61dane 2594 | . 2 ⊢ ((φ ∧ (A ≠ B ∧ C ≠ D)) → ψ) |
12 | 1, 2, 11 | pm2.61da2ne 2595 | 1 ⊢ (φ → ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 df-ne 2518 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |