Theorem ceqsexg 3556

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		nfe1 2088	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝐴 ∧ 𝜑)
2		ceqsexg.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfbi 1867	. 2 ⊢ Ⅎ𝑥(∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
4		ceqex 3555	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5		ceqsexg.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	4, 5	bibi12d 338	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜑) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)))
7		biid 253	. 2 ⊢ (𝜑 ↔ 𝜑)
8	3, 6, 7	vtoclg1f 3480	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508  ∃wex 1743  Ⅎwnf 1747   ∈ wcel 2051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-ext 2745

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-v 3412

This theorem is referenced by:  ceqsexgv  3557