Theorem eqsb3 2244

Description: Substitution applied to an atomic wff (class version of equsb3 1925). (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqsb3lem 2243	. . 3 ⊢ ([𝑤 / 𝑥]𝑥 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1739	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑤]𝑤 = 𝐴)
3		nfv 1509	. . 3 ⊢ Ⅎ𝑤 𝑥 = 𝐴
4	3	sbco2 1939	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑥]𝑥 = 𝐴)
5		eqsb3lem 2243	. 2 ⊢ ([𝑦 / 𝑤]𝑤 = 𝐴 ↔ 𝑦 = 𝐴)
6	2, 4, 5	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Syntax hints:   ↔ wb 104   = wceq 1332  [wsb 1736

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-cleq 2133

This theorem is referenced by:  pm13.183  2826  eqsbc3  2952