Theorem ceqsal 1363

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis.

Hypotheses

Ref	Expression
ceqsal.1	⊢ (ψ → ∀xψ)
ceqsal.2	⊢ A ∈ V
ceqsal.3	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ceqsal	⊢ (∀x(x = A → φ) ↔ ψ)

Distinct variable group(s):   x,A

Proof of Theorem ceqsal

Step	Hyp	Ref	Expression
1		ceqsal.2	. 2 ⊢ A ∈ V
2		ceqsal.1	. . 3 ⊢ (ψ → ∀xψ)
3		ceqsal.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
4	2, 3	ceqsalg 1362	. 2 ⊢ (A ∈ V → (∀x(x = A → φ) ↔ ψ))
5	1, 4	ax-mp 6	1 ⊢ (∀x(x = A → φ) ↔ ψ)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  ceqsalv 1364  sbc6 1453

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-gen 677  ax-9 799  ax-12 802  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

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