Theorem ceqsalg 1822

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:   x,A

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		ceqsalg.2	. . . . . . 7 ⊢ (x = A → (φ ↔ ψ))
2	1	biimpd 153	. . . . . 6 ⊢ (x = A → (φ → ψ))
3	2	a2i 9	. . . . 5 ⊢ ((x = A → φ) → (x = A → ψ))
4	3	19.20i 991	. . . 4 ⊢ (∀x(x = A → φ) → ∀x(x = A → ψ))
5		ceqsalg.1	. . . . 5 ⊢ (ψ → ∀xψ)
6	5	19.23 1062	. . . 4 ⊢ (∀x(x = A → ψ) ↔ (∃x x = A → ψ))
7	4, 6	sylib 198	. . 3 ⊢ (∀x(x = A → φ) → (∃x x = A → ψ))
8		elex 1816	. . 3 ⊢ (A ∈ B → ∃x x = A)
9	7, 8	syl5com 52	. 2 ⊢ (A ∈ B → (∀x(x = A → φ) → ψ))
10	1	biimprcd 156	. . 3 ⊢ (ψ → (x = A → φ))
11	5, 10	19.21ai 997	. 2 ⊢ (ψ → ∀x(x = A → φ))
12	9, 11	impbid1 516	1 ⊢ (A ∈ B → (∀x(x = A → φ) ↔ ψ))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 953   = wceq 955   ∈ wcel 957  ∃wex 979

This theorem is referenced by:  ceqsal 1823  sbc6g 1952  sucprcreg 4583  spwpr2 8615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  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

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