Theorem equsb3 2432

Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2034. (Revised by Wolf Lammen, 21-Sep-2018.)

Assertion

Ref	Expression
equsb3	⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3lem 2431	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧)
2	1	sbbii 1887	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3		sbcom3 2411	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧)
4		nfv 1843	. . . 4 ⊢ Ⅎ𝑤[𝑥 / 𝑦]𝑦 = 𝑧
5	4	sbf 2380	. . 3 ⊢ ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
6	3, 5	bitri 264	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
7		equsb3lem 2431	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧)
8	2, 6, 7	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-10 2019  ax-12 2047  ax-13 2246

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

This theorem is referenced by:  sb8eu  2503  mo3  2507  sb8iota  5858  mo5f  29324  mptsnunlem  33185  wl-equsb3  33337  wl-mo3t  33358  wl-sb8eut  33359  frege55lem1b  38189  sbeqal1  38598