Theorem bj-clelsb3 32973

Description: Remove dependency on ax-ext 2631 (and df-cleq 2644) from clelsb3 2758. (Contributed by BJ, 24-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-clelsb3	⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bj-clelsb3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1883	. . 3 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
2	1	sbco2 2443	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑧]𝑧 ∈ 𝐴)
3		nfv 1883	. . . 4 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐴
4		eleq1w 2713	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	3, 4	sbie 2436	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	sbbii 1944	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
7		nfv 1883	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
8		eleq1w 2713	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9	7, 8	sbie 2436	. 2 ⊢ ([𝑥 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
10	2, 6, 9	3bitr3i 290	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 196  [wsb 1937   ∈ wcel 2030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clel 2647

This theorem is referenced by:  bj-hblem  32974