Theorem rabxfr 5365

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 1554	. 2 ⊢ ⊤
2		rabxfr.1	. . 3 ⊢ Ⅎ𝑦𝐵
3		rabxfr.2	. . 3 ⊢ Ⅎ𝑦𝐶
4		rabxfr.3	. . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)
5	4	adantl 484	. . 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 5364	. 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
9	1, 8	mpan 698	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1550  ⊤wtru 1551   ∈ wcel 2132  Ⅎwnfc 2899  {crab 3404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1553  df-ex 1790  df-nf 1794  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-rab 3405  df-v 3446

This theorem is referenced by: (None)