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Mirrors > Home > NFE Home > Th. List > syl6eqel | GIF version |
Description: A membership and equality inference. (Contributed by NM, 4-Jan-2006.) |
Ref | Expression |
---|---|
syl6eqel.1 | ⊢ (φ → A = B) |
syl6eqel.2 | ⊢ B ∈ C |
Ref | Expression |
---|---|
syl6eqel | ⊢ (φ → A ∈ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6eqel.1 | . 2 ⊢ (φ → A = B) | |
2 | syl6eqel.2 | . . 3 ⊢ B ∈ C | |
3 | 2 | a1i 10 | . 2 ⊢ (φ → B ∈ C) |
4 | 1, 3 | eqeltrd 2427 | 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: syl6eqelr 2442 snex 4111 sikss1c1c 4267 ins2kss 4279 ins3kss 4280 iotaex 4356 eladdc 4398 ssfin 4470 f0cli 5418 elimdelov 5573 ndmovcl 5614 brfns 5833 muccl 6132 ncaddccl 6144 ceclb 6183 cet 6234 nclenn 6249 nchoicelem12 6300 nchoicelem17 6305 nchoicelem18 6306 nchoicelem19 6307 |
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