Theorem ceqsexv2d 3533

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 3498	. 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 2108  Vcvv 3480

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 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2816

This theorem is referenced by:  en0  9058  en0r  9060  ensn1  9061  0domg  9140  tz9.1  9769  cplem2  9930  karden  9935  pwmnd  18950  2lgslem1  27438  griedg0prc  29281  1loopgrvd2  29521  bnj150  34890  fnchoice  45034  nfermltl8rev  47729  nfermltl2rev  47730  nfermltlrev  47731  nn0mnd  48095  rrx2xpreen  48640