Theorem eloprabg 6914

Description: The law of concretion for operation class abstraction. Compare elopab 5133. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
eloprabg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
eloprabg.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
eloprabg.3	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
eloprabg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem eloprabg

Step	Hyp	Ref	Expression
1		eloprabg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		eloprabg.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3		eloprabg.3	. . 3 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
4	1, 2, 3	syl3an9b 1546	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜃))
5	4	eloprabga 6913	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ⟨cop 4327  {coprab 6815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-oprab 6818

This theorem is referenced by:  ov  6946  ovg  6965  brbtwn  25999  isnvlem  27795  isphg  28002  fvtransport  32466  brcolinear2  32492  colineardim1  32495  fvray  32575  fvline  32578