Theorem sbc8g 2765

Description: This is the closest we can get to df-sbc 2759 if we start from dfsbcq 2760 (see its comments) and dfsbcq2 2761. (Contributed by NM, 18-Nov-2008.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc8g	⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))

Proof of Theorem sbc8g

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 2760	. 2 ⊢ (y = A → ([y / x]φ ↔ [A / x]φ))
2		eleq1 2097	. 2 ⊢ (y = A → (y ∈ {x ∣ φ} ↔ A ∈ {x ∣ φ}))
3		df-clab 2024	. . 3 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
4		equid 1586	. . . 4 ⊢ y = y
5		dfsbcq2 2761	. . . 4 ⊢ (y = y → ([y / x]φ ↔ [y / x]φ))
6	4, 5	ax-mp 7	. . 3 ⊢ ([y / x]φ ↔ [y / x]φ)
7	3, 6	bitr2i 174	. 2 ⊢ ([y / x]φ ↔ y ∈ {x ∣ φ})
8	1, 2, 7	vtoclbg 2608	1 ⊢ (A ∈ 𝑉 → ([A / x]φ ↔ A ∈ {x ∣ φ}))

Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  [wsb 1642  {cab 2023  [wsbc 2758

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759

This theorem is referenced by:  bj-elssuniab  9199