Theorem eqsb1 2300

Description: Substitution for the left-hand side in an equality. Class version of equsb3 1970. (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 2299	. . 3 ⊢ ([𝑤 / 𝑥]𝑥 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 1779	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑤]𝑤 = 𝐴)
3		nfv 1542	. . 3 ⊢ Ⅎ𝑤 𝑥 = 𝐴
4	3	sbco2 1984	. 2 ⊢ ([𝑦 / 𝑤][𝑤 / 𝑥]𝑥 = 𝐴 ↔ [𝑦 / 𝑥]𝑥 = 𝐴)
5		eqsb1lem 2299	. 2 ⊢ ([𝑦 / 𝑤]𝑤 = 𝐴 ↔ 𝑦 = 𝐴)
6	2, 4, 5	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)

Syntax hints:   ↔ wb 105   = wceq 1364  [wsb 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-cleq 2189

This theorem is referenced by:  pm13.183  2902  eqsbc1  3029