Theorem syl5eqel 2121

Description: B membership and equality inference. (Contributed by NM, 4-Jan-2006.)

Hypotheses

Ref	Expression
syl5eqel.1	⊢ A = B
syl5eqel.2	⊢ (φ → B ∈ 𝐶)

Assertion

Ref	Expression
syl5eqel	⊢ (φ → A ∈ 𝐶)

Proof of Theorem syl5eqel

Step	Hyp	Ref	Expression
1		syl5eqel.1	. . 3 ⊢ A = B
2	1	a1i 9	. 2 ⊢ (φ → A = B)
3		syl5eqel.2	. 2 ⊢ (φ → B ∈ 𝐶)
4	2, 3	eqeltrd 2111	1 ⊢ (φ → A ∈ 𝐶)

Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030  df-clel 2033

This theorem is referenced by:  syl5eqelr  2122  csbexga  3876  otexg  3958  tpexg  4145  dmresexg  4577  f1oabexg  5081  funfvex  5135  riotaexg  5415  riotaprop  5434  fnovex  5481  fovrn  5585  fnovrn  5590  cofunexg  5680  cofunex2g  5681  abrexex2g  5689  xpexgALT  5702  mpt2fvex  5771  tfrex  5895  frec0g  5922  ecexg  6046  qsexg  6098  negcl  7008  frecuzrdgrrn  8875  bj-snexg  9367