Theorem rabxfr 4386

Description: Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (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 1335	. 2 ⊢ ⊤
2		rabxfr.1	. . 3 ⊢ Ⅎ𝑦𝐵
3		rabxfr.2	. . 3 ⊢ Ⅎ𝑦𝐶
4		rabxfr.3	. . . 4 ⊢ (𝑦 ∈ 𝐷 → 𝐴 ∈ 𝐷)
5	4	adantl 275	. . 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 4385	. 2 ⊢ ((⊤ ∧ 𝐵 ∈ 𝐷) → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))
9	1, 8	mpan 420	1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ {𝑥 ∈ 𝐷 ∣ 𝜑} ↔ 𝐵 ∈ {𝑦 ∈ 𝐷 ∣ 𝜓}))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  ⊤wtru 1332   ∈ wcel 1480  Ⅎwnfc 2266  {crab 2418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683

This theorem is referenced by: (None)