Theorem ceqsalg 2714

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 2700	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		nfa1 1521	. . . 4 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝐴 → 𝜑)
3		ceqsalg.1	. . . 4 ⊢ Ⅎ𝑥𝜓
4		ceqsalg.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	4	biimpd 143	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
6	5	a2i 11	. . . . 5 ⊢ ((𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
7	6	sps 1517	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜑) → (𝑥 = 𝐴 → 𝜓))
8	2, 3, 7	exlimd 1576	. . 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 159	. . 3 ⊢ (𝜓 → (𝑥 = 𝐴 → 𝜑))
11	3, 10	alrimi 1502	. 2 ⊢ (𝜓 → ∀𝑥(𝑥 = 𝐴 → 𝜑))
12	9, 11	impbid1 141	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝑥 = 𝐴 → 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436  ∃wex 1468   ∈ wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688

This theorem is referenced by:  ceqsal  2715  sbc6g  2933  uniiunlem  3185  sucprcreg  4464  funimass4  5472  ralrnmpo  5885