Theorem rabidim2 43791

Description: Membership in a restricted abstraction, implication. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
rabidim2	⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

Proof of Theorem rabidim2

Step	Hyp	Ref	Expression
1		rabid 3453	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	simprbi 498	1 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2107  {crab 3433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434

This theorem is referenced by:  infnsuprnmpt  43954  preimagelt  45415  preimalegt  45416  pimrecltpos  45424  pimiooltgt  45426  pimrecltneg  45440  smfaddlem1  45479  smflimlem2  45488  smfrec  45505  smfmullem4  45510  smfdiv  45513  smfsupxr  45532  smfinflem  45533  smflimsuplem7  45542  smflimsuplem8  45543  fsupdm  45558  finfdm  45562