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Mirrors > Home > NFE Home > Th. List > syl5eleq | GIF version |
Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
syl5eleq.1 | ⊢ A ∈ B |
syl5eleq.2 | ⊢ (φ → B = C) |
Ref | Expression |
---|---|
syl5eleq | ⊢ (φ → A ∈ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5eleq.1 | . . 3 ⊢ A ∈ B | |
2 | 1 | a1i 10 | . 2 ⊢ (φ → A ∈ B) |
3 | syl5eleq.2 | . 2 ⊢ (φ → B = C) | |
4 | 2, 3 | eleqtrd 2429 | 1 ⊢ (φ → A ∈ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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-an 360 df-ex 1542 df-cleq 2346 df-clel 2349 |
This theorem is referenced by: syl5eleqr 2440 enadj 6060 |
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