Theorem opelopabgf 5429

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 5427 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 5419	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥𝐵
3		opelopabgf.x	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfsbcw 3796	. . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓
5		opelopabgf.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	sbcbidv 3829	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
7	4, 6	sbciegf 3811	. . 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 3811	. . 3 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
11	7, 10	sylan9bb 512	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜒))
12	1, 11	syl5bb 285	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  [wsbc 3774  ⟨cop 4575  {copab 5130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-opab 5131

This theorem is referenced by:  oprabv  7216