Theorem opelopabaf 4368

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4366 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 4354	. 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 1576	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
7		opelopabaf.3	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
8	4, 5, 6, 7	sbc2iegf 3102	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
9	2, 3, 8	mp2an 426	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
10	1, 9	bitri 184	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397  Ⅎwnf 1508   ∈ wcel 2202  Vcvv 2802  [wsbc 3031  ⟨cop 3672  {copab 4149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151

This theorem is referenced by: (None)