Theorem csbief 2029

Description: Conversion of implicit substitution to explicit substitution into a class.

Hypotheses

Ref	Expression
csbief.1	⊢ A ∈ V
csbief.2	⊢ (y ∈ C → ∀x y ∈ C)
csbief.3	⊢ (x = A → B = C)

Assertion

Ref	Expression
csbief	⊢ [A / x]B = C

Distinct variable groups:   x,A   y,C   x,y

Proof of Theorem csbief

Step	Hyp	Ref	Expression
1		csbief.3	. . 3 ⊢ (x = A → B = C)
2	1	ax-gen 962	. 2 ⊢ ∀x(x = A → B = C)
3		csbief.1	. . 3 ⊢ A ∈ V
4		csbief.2	. . 3 ⊢ (y ∈ C → ∀x y ∈ C)
5	3, 4	csbieb 2027	. 2 ⊢ (∀x(x = A → B = C) ↔ [A / x]B = C)
6	2, 5	mpbi 189	1 ⊢ [A / x]B = C

Syntax hints:   → wi 3  ∀wal 953   = wceq 955   ∈ wcel 957  Vcvv 1808  [csb 1998

This theorem is referenced by:  eqerlem 4263  binomlem1 7019  binomlem2 7020  binomlem4 7022  iserzshft2 7060

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-v 1809  df-sbc 1939  df-csb 1999

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