Theorem opelopabg 5490

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopabg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopabg.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
opelopabg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

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

Proof of Theorem opelopabg

Step	Hyp	Ref	Expression
1		opelopabg.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2		opelopabg.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
3	1, 2	sylan9bb 509	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒))
4	3	opelopabga 5485	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  {copab 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149

This theorem is referenced by:  opelopab  5494  fvopab3g  6940  fvopab3ig  6941  ov  7508  ovg  7529  joindef  18337  meetdef  18351  isvclem  30645  adj1  32001  adjeq  32003  linedegen  36322  bj-finsumval0  37596  opelopab3  38036  isdivrngo  38268  iscom2  38313  inxprnres  38616  dihpN  41779  tfsconcatfv2  43765