Theorem opelopabaf 4010

Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4008 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
opelopabaf.x	⊢ Ⅎ𝑥𝜓
opelopabaf.y	⊢ Ⅎ𝑦𝜓
opelopabaf.1	⊢ 𝐴 ∈ V
opelopabaf.2	⊢ 𝐵 ∈ V
opelopabaf.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

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

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

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

Proof of Theorem opelopabaf

Step	Hyp	Ref	Expression
1		opelopabsb 3997	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2		opelopabaf.1	. . 3 ⊢ 𝐴 ∈ V
3		opelopabaf.2	. . 3 ⊢ 𝐵 ∈ V
4		opelopabaf.x	. . . 4 ⊢ Ⅎ𝑥𝜓
5		opelopabaf.y	. . . 4 ⊢ Ⅎ𝑦𝜓
6		nfv 1421	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ V
7		opelopabaf.3	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
8	4, 5, 6, 7	sbc2iegf 2828	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
9	2, 3, 8	mp2an 402	. 2 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)
10	1, 9	bitri 173	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  Ⅎwnf 1349   ∈ wcel 1393  Vcvv 2557  [wsbc 2764  ⟨cop 3378  {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819

This theorem is referenced by: (None)