Theorem clelsb1f 2323

Description: Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2155). (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
clelsb1f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
clelsb1f	⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Proof of Theorem clelsb1f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb1f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2313	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3	2	sbco2 1965	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
4		nfv 1528	. . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
5		eleq1w 2238	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
6	4, 5	sbie 1791	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
7	6	sbbii 1765	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
8		nfv 1528	. . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
9		eleq1w 2238	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	8, 9	sbie 1791	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
11	3, 7, 10	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 105  [wsb 1762   ∈ wcel 2148  Ⅎwnfc 2306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308

This theorem is referenced by:  rmo3f  2936