Theorem sbc5 3071

Description: An equivalence for class substitution. (Contributed by NM, 23-Aug-1993.) (Revised by Mario Carneiro, 12-Oct-2016.)

Assertion

Ref	Expression
sbc5	⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ))

Distinct variable group:   x,A

Allowed substitution hint:   φ(x)

Proof of Theorem sbc5

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3056	. 2 ⊢ ([̣A / x]̣φ → A ∈ V)
2		exsimpl 1592	. . 3 ⊢ (∃x(x = A ∧ φ) → ∃x x = A)
3		isset 2864	. . 3 ⊢ (A ∈ V ↔ ∃x x = A)
4	2, 3	sylibr 203	. 2 ⊢ (∃x(x = A ∧ φ) → A ∈ V)
5		dfsbcq2 3050	. . 3 ⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ))
6		eqeq2 2362	. . . . 5 ⊢ (y = A → (x = y ↔ x = A))
7	6	anbi1d 685	. . . 4 ⊢ (y = A → ((x = y ∧ φ) ↔ (x = A ∧ φ)))
8	7	exbidv 1626	. . 3 ⊢ (y = A → (∃x(x = y ∧ φ) ↔ ∃x(x = A ∧ φ)))
9		sb5 2100	. . 3 ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ))
10	5, 8, 9	vtoclbg 2916	. 2 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ)))
11	1, 4, 10	pm5.21nii 342	1 ⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ))

This proof depends on syntax axioms:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  Vcvv 2860  [̣wsbc 3047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048

This theorem is used by:  sbc6g  3072  sbc7  3074  sbciegft  3077  sbccomlem  3117  csb2  3139  rexsns  3765