Theorem wl-clelsb3df 35028

Description: Deduction version of clelsb3f 2960. (Contributed by Wolf Lammen, 29-May-2023.)

Hypotheses

Ref	Expression
clelsb3df.1	⊢ Ⅎ𝑦𝜑
clelsb3df.2	⊢ (𝜑 → Ⅎ𝑦𝐴)

Assertion

Ref	Expression
wl-clelsb3df	⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))

Proof of Theorem wl-clelsb3df

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1915	. . 3 ⊢ Ⅎ𝑤𝜑
2		clelsb3df.1	. . 3 ⊢ Ⅎ𝑦𝜑
3		clelsb3df.2	. . . 4 ⊢ (𝜑 → Ⅎ𝑦𝐴)
4	3	nfcrd 2945	. . 3 ⊢ (𝜑 → Ⅎ𝑦 𝑤 ∈ 𝐴)
5	1, 2, 4	sbco2d 2531	. 2 ⊢ (𝜑 → ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴))
6		clelsb3 2917	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
7	6	sbbii 2081	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
8		clelsb3 2917	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
9	5, 7, 8	3bitr3g 316	1 ⊢ (𝜑 → ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  Ⅎwnfc 2936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clel 2870  df-nfc 2938

This theorem is referenced by:  wl-dfrabf  35029