Theorem ceqsexg 2970

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

Hypotheses

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

Assertion

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

Distinct variable group:   x,A

Allowed substitution hints:   φ(x)   ψ(x)   V(x)

Proof of Theorem ceqsexg

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲxA
2		nfe1 1732	. . 3 ⊢ Ⅎx∃x(x = A ∧ φ)
3		ceqsexg.1	. . 3 ⊢ Ⅎxψ
4	2, 3	nfbi 1834	. 2 ⊢ Ⅎx(∃x(x = A ∧ φ) ↔ ψ)
5		ceqex 2969	. . 3 ⊢ (x = A → (φ ↔ ∃x(x = A ∧ φ)))
6		ceqsexg.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	bibi12d 312	. 2 ⊢ (x = A → ((φ ↔ φ) ↔ (∃x(x = A ∧ φ) ↔ ψ)))
8		biid 227	. 2 ⊢ (φ ↔ φ)
9	1, 4, 7, 8	vtoclgf 2913	1 ⊢ (A ∈ V → (∃x(x = A ∧ φ) ↔ ψ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861

This theorem is referenced by:  ceqsexgv  2971