Theorem equsb3 1825

Description: Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)

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 1824	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧)
2	1	sbbii 1648	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
3		ax-17 1419	. . 3 ⊢ (𝑦 = 𝑧 → ∀𝑤 𝑦 = 𝑧)
4	3	sbco2v 1821	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
5		equsb3lem 1824	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧)
6	2, 4, 5	3bitr3i 199	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Syntax hints:   ↔ wb 98  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  sb8eu  1913  sb8euh  1923  sb8iota  4874