Theorem clelsb1 2866

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2114). (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 2821	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2		eleq1w 2821	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	1, 2	sbievw2 2099	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 205  [wsb 2067   ∈ wcel 2106

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 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clel 2816

This theorem is referenced by:  hblem  2870  hblemg  2871  abbi2dv  2877  nfcriiOLD  2900  clelsb1fw  2911  clelsb1f  2912  cbvreuwOLD  3377  cbvreu  3381  elrabi  3618  sbcel1v  3787  rmo3  3822  kmlem15  9920  iuninc  30900  measiuns  32185  ballotlemodife  32464  bj-nfcf  35111  ellimcabssub0  43158