Theorem eqsb3 2141

Description: Substitution applied to an atomic wff (class version of equsb3 1825). (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 2140	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1648	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3		nfv 1421	. . 3 ⊢ Ⅎ𝑤 𝑦 = 𝐴
4	3	sbco2 1839	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5		eqsb3lem 2140	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝐴 ↔ 𝑥 = 𝐴)
6	2, 4, 5	3bitr3i 199	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 98   = wceq 1243  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-cleq 2033

This theorem is referenced by:  pm13.183  2681  eqsbc3  2802