Theorem clelsb1 2343

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2216). (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		nfv 1581	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
2	1	sbco2 2025	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
3		nfv 1581	. . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
4		eleq1 2301	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	3, 4	sbie 1844	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
6	5	sbbii 1818	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
7		nfv 1581	. . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
8		eleq1 2301	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
9	7, 8	sbie 1844	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
10	2, 6, 9	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 105  [wsb 1815   ∈ wcel 2209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234

This theorem is used by:  hblem  2346  eqabdv  2369  nfraldya  2585  nfrexdya  2586  cbvreu  2784  sbcel1v  3114  rmo3  3144  setindel  4685  elirr  4688  en2lp  4701  zfregfr  4721  tfi  4729  bdcriota  17080