Theorem opelopabgf 5536

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5534 uses bound-variable hypotheses in place of distinct variable conditions. (Contributed by Alexander van der Vekens, 8-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
opelopabgf	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨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)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opelopabgf

Step	Hyp	Ref	Expression
1		opelopabsb 5526	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		nfcv 2892	. . . . 5 ⊢ Ⅎ𝑥𝐵
3		opelopabgf.x	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfsbcw 3790	. . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓
5		opelopabgf.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	sbcbidv 3828	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
7	4, 6	sbciegf 3808	. . 3 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
8		opelopabgf.y	. . . 4 ⊢ Ⅎ𝑦𝜒
9		opelopabgf.2	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
10	8, 9	sbciegf 3808	. . 3 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
11	7, 10	sylan9bb 508	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜒))
12	1, 11	bitrid 282	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533  Ⅎwnf 1777   ∈ wcel 2098  [wsbc 3768  ⟨cop 4630  {copab 5205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5206

This theorem is referenced by:  oprabv  7477