Theorem elrabrd 3655

Description: Deduction version of elrab 3652, just like elrabd 3654, but backwards direction. (Contributed by Thierry Arnoux, 15-Jan-2026.)

Hypotheses

Ref	Expression
elrabrd.1	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
elrabrd.2	⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})

Assertion

Ref	Expression
elrabrd	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥

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

Proof of Theorem elrabrd

Step	Hyp	Ref	Expression
1		elrabrd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓})
2		elrabrd.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜒))
3	2	elrab 3652	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))
4	1, 3	sylib 220	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝜒))
5	4	simprd 499	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {crab 3416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458

This theorem is referenced by:  plngcplem  28994  cycpmconjslem2  33337  selvply1rhmlemb  33818  selvply1rhm0  33825  extvfvvcl  33834  extvfvcl  33835  mplmulmvr  33838  evlextv  33841  mplvrpmrhm  33846  psrmonprod  33851  esplymhp  33867  esplyfv1  33868  esplyfval3  33871  esplyind  33874