Theorem opelopabf 5555

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5552 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 5540	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		opelopabf.1	. . 3 ⊢ 𝐴 ∈ V
3		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝐵
4		opelopabf.x	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfsbcw 3813	. . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓
6		opelopabf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	sbcbidv 3851	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
8	5, 7	sbciegf 3831	. . 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 3831	. . 3 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
14	10, 13	ax-mp 5	. 2 ⊢ ([𝐵 / 𝑦]𝜓 ↔ 𝜒)
15	1, 9, 14	3bitri 297	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  Vcvv 3478  [wsbc 3791  ⟨cop 4637  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211

This theorem is referenced by:  pofun  5615  fmptco  7149  fmptcof2  32674