Theorem ceqsexv2d 3492

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) Shorten, reduce dv conditions. (Revised by Wolf Lammen, 5-Jun-2025.) (Proof shortened by SN, 5-Jun-2025.)

Hypotheses

Ref	Expression
ceqsexv2d.1	⊢ 𝐴 ∈ V
ceqsexv2d.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsexv2d.3	⊢ 𝜓

Assertion

Ref	Expression
ceqsexv2d	⊢ ∃𝑥𝜑

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem ceqsexv2d

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	. . 3 ⊢ 𝐴 ∈ V
2	1	isseti 3459	. 2 ⊢ ∃𝑥 𝑥 = 𝐴
3		ceqsexv2d.3	. . 3 ⊢ 𝜓
4		ceqsexv2d.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
5	3, 4	mpbiri 258	. 2 ⊢ (𝑥 = 𝐴 → 𝜑)
6	2, 5	eximii 1839	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-clel 2812

This theorem is referenced by:  en0  8960  en0r  8962  ensn1  8963  0domg  9037  tz9.1  9643  cplem2  9807  karden  9812  pwmnd  18867  2lgslem1  27366  griedg0prc  29342  1loopgrvd2  29582  bnj150  35045  exeltr  36678  permaxsep  45326  permaxnul  45327  permaxpow  45328  permaxpr  45329  permaxun  45330  permaxinf2lem  45331  nregmodel  45336  fnchoice  45352  nfermltl8rev  48065  nfermltl2rev  48066  nfermltlrev  48067  gpg5edgnedg  48453  nn0mnd  48502  rrx2xpreen  49042