Theorem clelsb3f 2768

Description: Substitution applied to an atomic wff (class version of elsb3 2434). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)

Hypothesis

Ref	Expression
clelsb3f.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
clelsb3f	⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Proof of Theorem clelsb3f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2758	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
3	2	sbco2 2415	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)
4		nfv 1843	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
5		eleq1 2689	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	4, 5	sbie 2408	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
7	6	sbbii 1887	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
8		nfv 1843	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
9		eleq1 2689	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
10	8, 9	sbie 2408	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
11	3, 7, 10	3bitr3i 290	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 196  [wsb 1880   ∈ wcel 1990  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  rmo3f  29335  suppss2f  29439  fmptdF  29456  disjdsct  29480  esumpfinvalf  30138