Theorem ceqsalg 2627

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 29-Oct-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ceqsalg

Step	Hyp	Ref	Expression
1		elisset 2613	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		nfa1 1474	. . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑)
3		ceqsalg.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4		ceqsalg.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 142	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	5	a2i 11	. . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
7	6	sps 1470	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
8	2, 3, 7	exlimd 1528	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (∃𝑥 𝑥 = 𝐴 → 𝜓))
9	1, 8	syl5com 29	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) → 𝜓))
10	4	biimprcd 158	. . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
11	3, 10	alrimi 1455	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
12	9, 11	impbid1 140	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282   = wceq 1284  Ⅎwnf 1389  ∃wex 1421   ∈ wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  ceqsal  2628  sbc6g  2839  uniiunlem  3082  sucprcreg  4292  funimass4  5245  ralrnmpt2  5635