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  8959  en0r  8961  ensn1  8962  0domg  9036  tz9.1  9642  cplem2  9806  karden  9811  pwmnd  18866  2lgslem1  27365  griedg0prc  29320  1loopgrvd2  29560  bnj150  35013  permaxsep  45284  permaxnul  45285  permaxpow  45286  permaxpr  45287  permaxun  45288  permaxinf2lem  45289  nregmodel  45294  fnchoice  45310  nfermltl8rev  48024  nfermltl2rev  48025  nfermltlrev  48026  gpg5edgnedg  48412  nn0mnd  48461  rrx2xpreen  49001