Theorem ceqsexg 3648

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 2154	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝐴 ∧ 𝜑)
2		ceqsexg.1	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfbi 1904	. 2 ⊢ Ⅎ𝑥(∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
4		ceqex 3647	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)))
5		ceqsexg.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	4, 5	bibi12d 348	. 2 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝜑) ↔ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)))
7		biid 263	. 2 ⊢ (𝜑 ↔ 𝜑)
8	3, 6, 7	vtoclg1f 3568	1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-v 3498

This theorem is referenced by:  ceqsexgvOLD  3650