Theorem equsb3 2004

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

Assertion

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

Distinct variable group:   𝑥,𝑧

Proof of Theorem equsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equsb3lem 2003	. . 3 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
2	1	sbbii 1813	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑤]𝑤 = 𝑧)
3		ax-17 1574	. . 3 ⊢ (𝑥 = 𝑧 → ∀𝑤 𝑥 = 𝑧)
4	3	sbco2vh 1998	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑥]𝑥 = 𝑧)
5		equsb3lem 2003	. 2 ⊢ ([𝑦 / 𝑤]𝑤 = 𝑧 ↔ 𝑦 = 𝑧)
6	2, 4, 5	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)

Syntax hints:   ↔ wb 105  [wsb 1810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811

This theorem is referenced by:  sb8eu  2092  sb8euh  2102  sb8iota  5294