Theorem opelopabf 4252

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4249 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
opelopabf.x	⊢ Ⅎ𝑥𝜓
opelopabf.y	⊢ Ⅎ𝑦𝜒
opelopabf.1	⊢ 𝐴 ∈ V
opelopabf.2	⊢ 𝐵 ∈ V
opelopabf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopabf.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

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

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

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

Proof of Theorem opelopabf

Step	Hyp	Ref	Expression
1		opelopabsb 4238	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		opelopabf.1	. . 3 ⊢ 𝐴 ∈ V
3		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		opelopabf.x	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfsbc 2971	. . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓
6		opelopabf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	sbcbidv 3009	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
8	5, 7	sbciegf 2982	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
9	2, 8	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓)
10		opelopabf.2	. . 3 ⊢ 𝐵 ∈ V
11		opelopabf.y	. . . 4 ⊢ Ⅎ𝑦𝜒
12		opelopabf.4	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
13	11, 12	sbciegf 2982	. . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
14	10, 13	ax-mp 5	. 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒)
15	1, 9, 14	3bitri 205	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  Vcvv 2726  [wsbc 2951  ⟨cop 3579  {copab 4042

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044

This theorem is referenced by:  pofun  4290  fmptco  5651