Theorem opelopabaf 5529

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5527 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
opelopabaf.x	⊢ Ⅎ𝑥𝜓
opelopabaf.y	⊢ Ⅎ𝑦𝜓
opelopabaf.1	⊢ 𝐴 ∈ V
opelopabaf.2	⊢ 𝐵 ∈ V
opelopabaf.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
opelopabaf	⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

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

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

Proof of Theorem opelopabaf

Step	Hyp	Ref	Expression
1		opelopabsb 5515	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		opelopabaf.1	. . 3 ⊢ 𝐴 ∈ V
3		opelopabaf.2	. . 3 ⊢ 𝐵 ∈ V
4		opelopabaf.x	. . . 4 ⊢ Ⅎ𝑥𝜓
5		opelopabaf.y	. . . 4 ⊢ Ⅎ𝑦𝜓
6		nfv 1913	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
7		opelopabaf.3	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
8	4, 5, 6, 7	sbc2iegf 3845	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
9	2, 3, 8	mp2an 692	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
10	1, 9	bitri 275	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  Ⅎwnf 1782   ∈ wcel 2107  Vcvv 3463  [wsbc 3770  ⟨cop 4612  {copab 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5186

This theorem is referenced by: (None)