Theorem elrabrd 3652

Description: Deduction version of elrab 3649, just like elrabd 3651, 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 3649	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))
4	1, 3	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝜒))
5	4	simprd 500	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {crab 3415

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456

This theorem is used by:  plngcplem  29078  cycpmconjslem2  33484  selvply1rhmlemb  33918  selvply1rhm0  33925  extvfvvcl  33934  extvfvcl  33935  mplmulmvr  33938  evlextv  33941  mplvrpmrhm  33946  psrmonprod  33951  esplymhp  33967  esplyfv1  33968  esplyfval3  33971  esplyind  33974  nmuladdel  36712