Theorem eqsb1 2269

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1939. (Contributed by Rodolfo Medina, 28-Apr-2010.)

Assertion

Ref	Expression
eqsb1	⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqsb1lem 2268	. . 3 ⊢ ([𝑤 / 𝑥]𝑥 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1753	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑤]𝑤 = 𝐴)
3		nfv 1516	. . 3 ⊢ Ⅎ𝑤 𝑥 = 𝐴
4	3	sbco2 1953	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑥]𝑥 = 𝐴)
5		eqsb1lem 2268	. 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 1343  [wsb 1750

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158

This theorem is referenced by:  pm13.183  2863  eqsbc1  2989