Theorem eqsb3 2182

Description: Substitution applied to an atomic wff (class version of equsb3 1866). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

Ref	Expression
eqsb3	⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eqsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqsb3lem 2181	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1688	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3		nfv 1461	. . 3 ⊢ Ⅎ𝑤 𝑦 = 𝐴
4	3	sbco2 1880	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5		eqsb3lem 2181	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝐴 ↔ 𝑥 = 𝐴)
6	2, 4, 5	3bitr3i 208	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 103   = wceq 1284  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-cleq 2074

This theorem is referenced by:  pm13.183  2732  eqsbc3  2853