Theorem clelsb1 2890

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2151). (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 2846	. 2 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
2		eleq1w 2846	. 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3	1, 2	sbievw2 2133	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2096   ∈ wcel 2143

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clel 2838

This theorem is used by:  hblem  2894  hblemg  2895  eqabdv  2896  clelsb1fw  2929  clelsb1f  2930  cbvreu  3408  elrabi  3646  sbcel1v  3809  rmo3  3842  kmlem15  10153  iuninc  32914  measiuns  34616  ballotlemodife  34897  bj-nfcf  37586  ellimcabssub0  46361