Theorem clelsb1 2861

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2115). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb1	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2817	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2		eleq1w 2817	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	1, 2	sbievw2 2100	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 205  [wsb 2068   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clel 2811

This theorem is referenced by:  hblem  2866  hblemg  2867  eqabdv  2868  nfcriiOLD  2897  clelsb1fw  2908  clelsb1f  2909  cbvreuwOLD  3413  cbvreu  3425  elrabi  3677  sbcel1v  3848  rmo3  3883  kmlem15  10156  iuninc  31780  measiuns  33204  ballotlemodife  33485  bj-nfcf  35792  ellimcabssub0  44320