Theorem sbciegft 3077

Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3078.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbciegft	⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ ↔ ψ))

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem sbciegft

Step	Hyp	Ref	Expression
1		sbc5 3071	. . 3 ⊢ ([̣A / x]̣φ ↔ ∃x(x = A ∧ φ))
2		bi1 178	. . . . . . . 8 ⊢ ((φ ↔ ψ) → (φ → ψ))
3	2	imim2i 13	. . . . . . 7 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (φ → ψ)))
4	3	imp3a 420	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → ((x = A ∧ φ) → ψ))
5	4	alimi 1559	. . . . 5 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x((x = A ∧ φ) → ψ))
6		19.23t 1800	. . . . . 6 ⊢ (Ⅎxψ → (∀x((x = A ∧ φ) → ψ) ↔ (∃x(x = A ∧ φ) → ψ)))
7	6	biimpa 470	. . . . 5 ⊢ ((Ⅎxψ ∧ ∀x((x = A ∧ φ) → ψ)) → (∃x(x = A ∧ φ) → ψ))
8	5, 7	sylan2 460	. . . 4 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → (∃x(x = A ∧ φ) → ψ))
9	8	3adant1 973	. . 3 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → (∃x(x = A ∧ φ) → ψ))
10	1, 9	syl5bi 208	. 2 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ → ψ))
11		bi2 189	. . . . . . . 8 ⊢ ((φ ↔ ψ) → (ψ → φ))
12	11	imim2i 13	. . . . . . 7 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (ψ → φ)))
13	12	com23 72	. . . . . 6 ⊢ ((x = A → (φ ↔ ψ)) → (ψ → (x = A → φ)))
14	13	alimi 1559	. . . . 5 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(ψ → (x = A → φ)))
15		19.21t 1795	. . . . . 6 ⊢ (Ⅎxψ → (∀x(ψ → (x = A → φ)) ↔ (ψ → ∀x(x = A → φ))))
16	15	biimpa 470	. . . . 5 ⊢ ((Ⅎxψ ∧ ∀x(ψ → (x = A → φ))) → (ψ → ∀x(x = A → φ)))
17	14, 16	sylan2 460	. . . 4 ⊢ ((Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → (ψ → ∀x(x = A → φ)))
18	17	3adant1 973	. . 3 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → (ψ → ∀x(x = A → φ)))
19		sbc6g 3072	. . . 4 ⊢ (A ∈ V → ([̣A / x]̣φ ↔ ∀x(x = A → φ)))
20	19	3ad2ant1 976	. . 3 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ ↔ ∀x(x = A → φ)))
21	18, 20	sylibrd 225	. 2 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → (ψ → [̣A / x]̣φ))
22	10, 21	impbid 183	1 ⊢ ((A ∈ V ∧ Ⅎxψ ∧ ∀x(x = A → (φ ↔ ψ))) → ([̣A / x]̣φ ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  [̣wsbc 3047

This theorem was proved from 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 theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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 referenced by:  sbciegf  3078  sbciedf  3082