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Mirrors > Home > NFE Home > Th. List > eqnetrd | GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
eqnetrd.1 | ⊢ (φ → A = B) |
eqnetrd.2 | ⊢ (φ → B ≠ C) |
Ref | Expression |
---|---|
eqnetrd | ⊢ (φ → A ≠ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetrd.2 | . 2 ⊢ (φ → B ≠ C) | |
2 | eqnetrd.1 | . . 3 ⊢ (φ → A = B) | |
3 | 2 | neeq1d 2529 | . 2 ⊢ (φ → (A ≠ C ↔ B ≠ C)) |
4 | 1, 3 | mpbird 223 | 1 ⊢ (φ → A ≠ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-cleq 2346 df-ne 2518 |
This theorem is referenced by: eqnetrrd 2536 vfin1cltv 4547 nchoicelem12 6300 nchoicelem14 6302 nchoicelem17 6305 |
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