Theorem ceqsexg 2908

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 2350	. 2 ⊢ Ⅎ𝑥𝐴
2		nfe1 1520	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝐴 ∧ 𝜑)
3		ceqsexg.1	. . 3 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfbi 1613	. 2 ⊢ Ⅎ𝑥(∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
5		ceqex 2907	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
6		ceqsexg.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	5, 6	bibi12d 235	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜑) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)))
8		biid 171	. 2 ⊢ (𝜑 ↔ 𝜑)
9	1, 4, 7, 8	vtoclgf 2836	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  Ⅎwnf 1484  ∃wex 1516   ∈ wcel 2178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  ceqsexgv  2909