Theorem elabd 3576

Description: Explicit demonstration the class {𝑥 ∣ 𝜓} is not empty by the example 𝐴. (Contributed by RP, 12-Aug-2020.) (Revised by AV, 23-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
elabd	⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})

Distinct variable groups:   𝑥,𝐴   𝜒,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem elabd

Step	Hyp	Ref	Expression
1		elabd.2	. 2 ⊢ (𝜑 → 𝜒)
2		elabd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		elabd.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
4	3	elabg 3571	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
5	2, 4	syl 17	. 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝜒))
6	1, 5	mpbird 260	1 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜓})

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1542   ∈ wcel 2114  {cab 2716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811

This theorem is referenced by:  wfrlem15  8000  lubval  17712  glbval  17725  sursubmefmnd  18179  injsubmefmnd  18180  branmfn  30042  orvcval  31996  nosupfv  33554  sticksstones3  39732  rngunsnply  40592  hoidmvlelem1  43697  cfsetsnfsetf  44113