Theorem sbcie 1455

Description: Conversion of implicit substitution to explicit class substitution.

Hypotheses

Ref	Expression
sbcie.1	⊢ A ∈ V
sbcie.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
sbcie	⊢ ([A / x]φ ↔ ψ)

Distinct variable group(s):   x,A   ψ,x

Proof of Theorem sbcie

Step	Hyp	Ref	Expression
1		sbcie.1	. . . 4 ⊢ A ∈ V
2	1	sbc5 1452	. . 3 ⊢ ([A / x]φ ↔ ∃x(x = A ∧ φ))
3		sbcie.2	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
4	3	biimpa 324	. . . 4 ⊢ ((x = A ∧ φ) → ψ)
5	4	19.23aiv 952	. . 3 ⊢ (∃x(x = A ∧ φ) → ψ)
6	2, 5	sylbi 174	. 2 ⊢ ([A / x]φ → ψ)
7	3	biimprcd 138	. . . 4 ⊢ (ψ → (x = A → φ))
8	7	19.21aiv 943	. . 3 ⊢ (ψ → ∀x(x = A → φ))
9	1	sbc6 1453	. . 3 ⊢ ([A / x]φ ↔ ∀x(x = A → φ))
10	8, 9	sylibr 175	. 2 ⊢ (ψ → [A / x]φ)
11	6, 10	impbi 139	1 ⊢ ([A / x]φ ↔ ψ)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

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