Theorem ceqsexv2d 3488

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 3487	. . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)
5	4	biimpri 229	. 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
6		exsimpr 1855	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥𝜑)
7	1, 5, 6	mp2b 10	1 ⊢ ∃𝑥𝜑

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525  ∃wex 1765   ∈ wcel 2083  Vcvv 3440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-nf 1770  df-cleq 2790  df-clel 2865

This theorem is referenced by:  ensn1  8428  tz9.1  9024  cplem2  9172  karden  9177  2lgslem1  25656  griedg0prc  26733  1loopgrvd2  26972  bnj150  31760  fnchoice  40846  nfermltl8rev  43411  nfermltl2rev  43412  nfermltlrev  43413  rrx2xpreen  44209