Theorem clelsb3fw 2901

Description: Substitution applied to an atomic wff (class version of elsb3 2120). Version of clelsb3f 2902 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypothesis

Ref	Expression
clelsb3fw.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
clelsb3fw	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem clelsb3fw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3fw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2884	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3	2	sbco2v 2333	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
4		clelsb3 2858	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
5	4	sbbii 2084	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
6		clelsb3 2858	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
7	3, 5, 6	3bitr3i 304	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 209  [wsb 2072   ∈ wcel 2112  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-10 2143  ax-11 2160  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-nf 1792  df-sb 2073  df-clel 2809  df-nfc 2879

This theorem is referenced by:  rmo3f  3636  suppss2f  30647  fmptdF  30667  disjdsct  30709  esumpfinvalf  31710