Theorem clelsb3f 2283

Description: Substitution applied to an atomic wff (class version of elsb3 1949). (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	⊢ Ⅎ𝑥𝐴

Assertion

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

Proof of Theorem clelsb3f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2273	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3	2	sbco2 1936	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
4		nfv 1508	. . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
5		eleq1w 2198	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
6	4, 5	sbie 1764	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
7	6	sbbii 1738	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
8		nfv 1508	. . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
9		eleq1w 2198	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	8, 9	sbie 1764	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
11	3, 7, 10	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 104   ∈ wcel 1480  [wsb 1735  Ⅎwnfc 2266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133  df-nfc 2268

This theorem is referenced by:  rmo3f  2876