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Theorem axprimlem1 4088
 Description: Lemma for the primitive axioms. Primitive form of equality to a singleton. (Contributed by SF, 25-Mar-2015.)
Assertion
Ref Expression
axprimlem1 (a = {B} ↔ c(c ac = B))
Distinct variable groups:   a,c   B,c
Allowed substitution hint:   B(a)

Proof of Theorem axprimlem1
StepHypRef Expression
1 dfcleq 2347 . 2 (a = {B} ↔ c(c ac {B}))
2 elsn 3748 . . . 4 (c {B} ↔ c = B)
32bibi2i 304 . . 3 ((c ac {B}) ↔ (c ac = B))
43albii 1566 . 2 (c(c ac {B}) ↔ c(c ac = B))
51, 4bitri 240 1 (a = {B} ↔ c(c ac = B))
 Colors of variables: wff set class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {csn 3737 This theorem is referenced by:  axprimlem2  4089  axsiprim  4093  axtyplowerprim  4094  axins2prim  4095  axins3prim  4096  snex  4111 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-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sn 3741
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