Theorem ceqsexv2d 3526

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
ceqsexv2d	⊢ ∃𝑥𝜑

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv2d

Step	Hyp	Ref	Expression
1		ceqsexv2d.3	. 2 ⊢ 𝜓
2		ceqsexv2d.1	. . . 4 ⊢ 𝐴 ∈ V
3		ceqsexv2d.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	ceqsexv 3523	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
5	4	biimpri 227	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
6		exsimpr 1872	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
7	1, 5, 6	mp2b 10	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474

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 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-clel 2810

This theorem is referenced by:  en0  8998  en0OLD  8999  en0r  9001  ensn1  9002  ensn1OLD  9003  0domg  9085  tz9.1  9708  cplem2  9869  karden  9874  pwmnd  18795  2lgslem1  26826  griedg0prc  28450  1loopgrvd2  28689  bnj150  33782  fnchoice  43548  nfermltl8rev  46246  nfermltl2rev  46247  nfermltlrev  46248  nn0mnd  46425  rrx2xpreen  47117