Theorem ceqsexg 1411

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

Hypotheses

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

Assertion

Ref	Expression
ceqsexg	⊢ (A ∈ B → (∃x(x = A ∧ φ) ↔ ψ))

Distinct variable group(s):   x,A

Proof of Theorem ceqsexg

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbe1 709	. . 3 ⊢ (∃x(x = A ∧ φ) → ∀x∃x(x = A ∧ φ))
3		ceqsexg.1	. . 3 ⊢ (ψ → ∀xψ)
4	2, 3	hbbi 705	. 2 ⊢ ((∃x(x = A ∧ φ) ↔ ψ) → ∀x(∃x(x = A ∧ φ) ↔ ψ))
5		ceqex 1410	. . 3 ⊢ (x = A → (φ ↔ ∃x(x = A ∧ φ)))
6		ceqsexg.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	bibi12d 477	. 2 ⊢ (x = A → ((φ ↔ φ) ↔ (∃x(x = A ∧ φ) ↔ ψ)))
8		pm4.2 148	. 2 ⊢ (φ ↔ φ)
9	1, 4, 7, 8	vtoclgf 1382	1 ⊢ (A ∈ B → (∃x(x = A ∧ φ) ↔ ψ))

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

This theorem is referenced by:  ceqsexgv 1412  sbc5g 1450

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

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