Theorem bj-clelsb3 32848

Description: Remove dependency on ax-ext 2602 (and df-cleq 2615) from clelsb3 2729. (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 1843	. . 3 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐴
2	1	sbco2 2415	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑧]𝑧 ∈ 𝐴)
3		nfv 1843	. . . 4 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐴
4		eleq1w 2684	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
5	3, 4	sbie 2408	. . 3 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	sbbii 1887	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
7		nfv 1843	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
8		eleq1w 2684	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
9	7, 8	sbie 2408	. 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 1880   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clel 2618

This theorem is referenced by:  bj-hblem  32849