Theorem elsb3 2434

Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb3	⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)

Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝑧
2	1	sbco2 2415	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝑧)
3		nfv 1843	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝑧
4		elequ1 1997	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧))
5	3, 4	sbie 2408	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)
6	5	sbbii 1887	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝑧 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝑧)
7		nfv 1843	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝑧
8		elequ1 1997	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧))
9	7, 8	sbie 2408	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)
10	2, 6, 9	3bitr3i 290	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝑧 ↔ 𝑥 ∈ 𝑧)

Syntax hints:   ↔ wb 196  [wsb 1880

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-8 1992  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

This theorem is referenced by:  cvjust  2617