Theorem csb2 3059

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 3058	. 2 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵}
2		sbc5 2986	. . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵))
3	2	abbii 2293	. 2 ⊢ {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}
4	1, 3	eqtri 2198	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴 ∧ 𝑦 ∈ 𝐵)}

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  [wsbc 2962  ⦋csb 3057

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963  df-csb 3058

This theorem is referenced by: (None)