Theorem opelopabgf 4304

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopabg 4302 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 4294	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑥𝐵
3		opelopabgf.x	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfsbcw 3119	. . . 4 ⊢ Ⅎ𝑥[𝐵 / 𝑦]𝜓
5		opelopabgf.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
6	5	sbcbidv 3048	. . . 4 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ [𝐵 / 𝑦]𝜓))
7	4, 6	sbciegf 3021	. . 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 3021	. . 3 ⊢ (𝐵 ∈ 𝑊 → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
11	7, 10	sylan9bb 462	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜒))
12	1, 11	bitrid 192	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  [wsbc 2989  ⟨cop 3625  {copab 4093

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095

This theorem is referenced by:  opabfi  6999