Theorem eqsb3 2854

Description: Substitution applied to an atomic wff (class version of equsb3 2557). (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 2853	. . 3 ⊢ ([𝑤 / 𝑦]𝑦 = 𝐴 ↔ 𝑤 = 𝐴)
2	1	sbbii 2041	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑤]𝑤 = 𝐴)
3		nfv 1980	. . 3 ⊢ Ⅎ𝑤 𝑦 = 𝐴
4	3	sbco2 2540	. 2 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝐴 ↔ [𝑥 / 𝑦]𝑦 = 𝐴)
5		eqsb3lem 2853	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝐴 ↔ 𝑥 = 𝐴)
6	2, 4, 5	3bitr3i 290	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝐴 ↔ 𝑥 = 𝐴)

Syntax hints:   ↔ wb 196   = wceq 1620  [wsb 2034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-cleq 2741

This theorem is referenced by:  pm13.183  3472  eqsbc3  3604