Theorem clelsb1fw 2929

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2151). Version of clelsb1f 2930 with a disjoint variable condition, which does not require ax-13 2404. (Contributed by Rodolfo Medina, 28-Apr-2010.) Avoid ax-13 2404. (Revised by GG, 10-Jan-2024.)

Hypothesis

Ref	Expression
clelsb1fw.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
clelsb1fw	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem clelsb1fw

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb1fw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2917	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3	2	sbco2v 2364	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
4		clelsb1 2890	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
5	4	sbbii 2110	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
6		clelsb1 2890	. 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 2096   ∈ wcel 2143  Ⅎwnfc 2910

This theorem was proved from 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  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-nf 1814  df-sb 2097  df-clel 2838  df-nfc 2912

This theorem is referenced by:  rmo3f  3697  suppss2f  32983  fmptdF  33001  disjdsct  33048  esumpfinvalf  34466