Theorem ceqsalg 1362

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

Hypotheses

Ref	Expression
ceqsalg.1	⊢ (ψ → ∀xψ)
ceqsalg.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
ceqsalg	⊢ (A ∈ B → (∀x(x = A → φ) ↔ ψ))

Distinct variable group(s):   x,A

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.2	. . . . . . . 8 ⊢ (x = A → (φ ↔ ψ))
2	1	biimpd 135	. . . . . . 7 ⊢ (x = A → (φ → ψ))
3	2	a2i 8	. . . . . 6 ⊢ ((x = A → φ) → (x = A → ψ))
4	3	19.20i 691	. . . . 5 ⊢ (∀x(x = A → φ) → ∀x(x = A → ψ))
5		ceqsalg.1	. . . . . 6 ⊢ (ψ → ∀xψ)
6	5	19.23 745	. . . . 5 ⊢ (∀x(x = A → ψ) ↔ (∃x x = A → ψ))
7	4, 6	sylib 173	. . . 4 ⊢ (∀x(x = A → φ) → (∃x x = A → ψ))
8		elex 1356	. . . 4 ⊢ (A ∈ B → ∃x x = A)
9	7, 8	syl5 22	. . 3 ⊢ (∀x(x = A → φ) → (A ∈ B → ψ))
10	9	com12 13	. 2 ⊢ (A ∈ B → (∀x(x = A → φ) → ψ))
11	1	biimprcd 138	. . . 4 ⊢ (ψ → (x = A → φ))
12	5, 11	19.21ai 740	. . 3 ⊢ (ψ → ∀x(x = A → φ))
13	12	a1i 7	. 2 ⊢ (A ∈ B → (ψ → ∀x(x = A → φ)))
14	10, 13	impbid 397	1 ⊢ (A ∈ B → (∀x(x = A → φ) ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  ceqsal 1363  sbc6g 1451  sucprcreg 3451

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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