Theorem elrabrd 32576

Description: Deduction version of elrab 3647, just like elrabd 3649, 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 3647	. . 3 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜓} ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))
4	1, 3	sylib 218	. 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝜒))
5	4	simprd 495	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443

This theorem is referenced by:  extvfvvcl  33702  extvfvcl  33703  mplmulmvr  33706  evlextv  33709  mplvrpmrhm  33714  esplymhp  33728  esplyfv1  33729  esplyfval3  33732  esplyind  33733