Theorem ceqsexv2d 3502

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 3468	. 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 1837	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2804

This theorem is referenced by:  en0  8992  en0r  8994  ensn1  8995  0domg  9074  tz9.1  9689  cplem2  9850  karden  9855  pwmnd  18871  2lgslem1  27312  griedg0prc  29198  1loopgrvd2  29438  bnj150  34873  permaxsep  45004  permaxnul  45005  permaxpow  45006  permaxpr  45007  permaxun  45008  permaxinf2lem  45009  nregmodel  45014  fnchoice  45030  nfermltl8rev  47747  nfermltl2rev  47748  nfermltlrev  47749  nn0mnd  48171  rrx2xpreen  48712