Theorem eqsb3 2192

Description: Substitution applied to an atomic wff (class version of equsb3 1874). (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 2191	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1696	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3		nfv 1467	. . 3 ⊢ Ⅎ𝑤 𝑦 = 𝐴
4	3	sbco2 1888	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5		eqsb3lem 2191	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝐴 ↔ 𝑥 = 𝐴)
6	2, 4, 5	3bitr3i 209	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1290  [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082

This theorem is referenced by:  pm13.183  2757  eqsbc3  2881