Theorem equsb3 1922

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 1921	. . 3 ⊢ ([𝑤 / 𝑥]𝑥 = 𝑧 ↔ 𝑤 = 𝑧)
2	1	sbbii 1738	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑤]𝑤 = 𝑧)
3		ax-17 1506	. . 3 ⊢ (𝑥 = 𝑧 → ∀𝑤 𝑥 = 𝑧)
4	3	sbco2vh 1916	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝑧 ↔ [𝑦 / 𝑥]𝑥 = 𝑧)
5		equsb3lem 1921	. 2 ⊢ ([𝑦 / 𝑤]𝑤 = 𝑧 ↔ 𝑦 = 𝑧)
6	2, 4, 5	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑧 ↔ 𝑦 = 𝑧)

Syntax hints:   ↔ wb 104  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sb8eu  2010  sb8euh  2020  sb8iota  5090