Theorem rabxfr 5389

Description: Membership in a restricted class abstraction after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the formula defining the class abstraction. (Contributed by NM, 10-Jun-2005.)

Hypotheses

Ref	Expression
rabxfr.1	⊢ Ⅎ𝑦𝐵
rabxfr.2	⊢ Ⅎ𝑦𝐶
rabxfr.3	⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)
rabxfr.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
rabxfr.5	⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)

Assertion

Ref	Expression
rabxfr	⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐷   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rabxfr

Step	Hyp	Ref	Expression
1		tru 1572	. 2 ⊢ ⊤
2		rabxfr.1	. . 3 ⊢ Ⅎ𝑦𝐵
3		rabxfr.2	. . 3 ⊢ Ⅎ𝑦𝐶
4		rabxfr.3	. . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)
5	4	adantl 486	. . 3 ⊢ ((⊤ ∧ 𝑦 ∈ 𝐷) → 𝐴 ∈ 𝐷)
6		rabxfr.4	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7		rabxfr.5	. . 3 ⊢ (𝑦 = 𝐵 → 𝐴 = 𝐶)
8	2, 3, 5, 6, 7	rabxfrd 5388	. 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
9	1, 8	mpan 702	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568  ⊤wtru 1569   ∈ wcel 2141  Ⅎwnfc 2908  {crab 3414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3415  df-v 3455

This theorem is referenced by: (None)