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Mirrors > Home > ILE 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 9 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶) |
4 | 1, 3 | eqeltrd 2194 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 |
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-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-cleq 2110 df-clel 2113 |
This theorem is referenced by: syl6eqelr 2209 snexprc 4080 onsucelsucexmidlem 4414 dcextest 4465 nnpredcl 4506 ovprc 5774 nnmcl 6345 xpsnen 6683 xpfi 6786 snexxph 6806 ctssdclemn0 6963 exmidonfinlem 7017 indpi 7118 nq0m0r 7232 genpelxp 7287 un0mulcl 8979 znegcl 9053 zeo 9124 eqreznegel 9374 xnegcl 9583 modqid0 10091 q2txmodxeq0 10125 ser0 10255 expcllem 10272 m1expcl2 10283 bcval 10463 bccl 10481 hashinfom 10492 resqrexlemlo 10753 iserge0 11080 sumrbdclem 11113 fsum3cvg 11114 summodclem3 11117 summodclem2a 11118 fisumss 11129 binom 11221 bcxmas 11226 gcdval 11575 gcdcl 11582 lcmcl 11680 ssblps 12521 ssbl 12522 xmeter 12532 blssioo 12641 nninfsellemeqinf 13139 nninffeq 13143 isomninnlem 13152 trilpolemclim 13156 |
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