Theorem elrabsf 3784

Description: Membership in a restricted class abstraction, expressed with explicit class substitution. (The variation elrabf 3642 has implicit substitution). The hypothesis specifies that 𝑥 must not be a free variable in 𝐵. (Contributed by NM, 30-Sep-2003.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
elrabsf.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
elrabsf	⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))

Proof of Theorem elrabsf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq 3741	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2		elrabsf.1	. . 3 ⊢ Ⅎ𝑥𝐵
3		nfcv 2923	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfv 1947	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfsbc1v 3759	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
6		sbceq1a 3750	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
7	2, 3, 4, 5, 6	cbvrabw 3447	. 2 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑦 ∈ 𝐵 ∣ [𝑦 / 𝑥]𝜑}
8	1, 7	elrab2 3649	1 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ 𝜑} ↔ (𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Ⅎwnfc 2908  {crab 3413  [wsbc 3739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-sbc 3740

This theorem is used by:  frpoinsg  6346  onminesb  7807  tfisg  7865  mpoxopovel  8237  frinsg  9755  ac6num  10557  hashrabsn1  14518  bnj23  35349  bnj1204  35642  weiunlem  37251  rabrenfdioph  43820