Theorem csb2 3849

Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.)

Assertion

Ref	Expression
csb2	⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csb2

Step	Hyp	Ref	Expression
1		df-csb 3848	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbc5 3767	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵))
3	2	abbii 2828	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}
4	1, 3	eqtri 2784	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739  [wsbc 3739  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-sbc 3740  df-csb 3848

This theorem is used by: (None)