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 2822	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2		eleq1w 2822	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	1, 2	sbievw2 2096	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 206  [wsb 2062   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-sb 2063  df-clel 2814

This theorem is referenced by:  hblem  2871  hblemg  2872  eqabdv  2873  clelsb1fw  2907  clelsb1f  2908  cbvreuwOLD  3413  cbvreu  3425  elrabi  3690  sbcel1v  3862  rmo3  3898  kmlem15  10203  iuninc  32581  measiuns  34198  ballotlemodife  34479  bj-nfcf  36906  ellimcabssub0  45573