Theorem clelsb3 2917

Description: Substitution applied to an atomic wff (class version of elsb3 2119). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
clelsb3	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2872	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2		eleq1w 2872	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	1, 2	sbievw2 2104	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 209  [wsb 2069   ∈ wcel 2111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clel 2870

This theorem is referenced by:  hblem  2920  hblemg  2921  abbi2dv  2927  nfcriiOLD  2949  clelsb3fw  2959  clelsb3f  2960  cbvreuw  3389  cbvreu  3394  sbcel1v  3786  rmo3  3818  kmlem15  9575  iuninc  30324  measiuns  31586  ballotlemodife  31865  bj-nfcf  34366  wl-dfrabv  35027  wl-clelsb3df  35028  wl-dfrabf  35029  ellimcabssub0  42259