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Theorem cbvopab1 4875
Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
cbvopab1.1 𝑧𝜑
cbvopab1.2 𝑥𝜓
cbvopab1.3 (𝑥 = 𝑧 → (𝜑𝜓))
Assertion
Ref Expression
cbvopab1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Distinct variable groups:   𝑥,𝑦   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem cbvopab1
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1992 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2 nfv 1992 . . . . . . 7 𝑥 𝑤 = ⟨𝑣, 𝑦
3 nfs1v 2574 . . . . . . 7 𝑥[𝑣 / 𝑥]𝜑
42, 3nfan 1977 . . . . . 6 𝑥(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
54nfex 2301 . . . . 5 𝑥𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
6 opeq1 4553 . . . . . . . 8 (𝑥 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑦⟩)
76eqeq2d 2770 . . . . . . 7 (𝑥 = 𝑣 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑣, 𝑦⟩))
8 sbequ12 2258 . . . . . . 7 (𝑥 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑥]𝜑))
97, 8anbi12d 749 . . . . . 6 (𝑥 = 𝑣 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
109exbidv 1999 . . . . 5 (𝑥 = 𝑣 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
111, 5, 10cbvex 2417 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑))
12 nfv 1992 . . . . . . 7 𝑧 𝑤 = ⟨𝑣, 𝑦
13 cbvopab1.1 . . . . . . . 8 𝑧𝜑
1413nfsb 2577 . . . . . . 7 𝑧[𝑣 / 𝑥]𝜑
1512, 14nfan 1977 . . . . . 6 𝑧(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
1615nfex 2301 . . . . 5 𝑧𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
17 nfv 1992 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)
18 opeq1 4553 . . . . . . . 8 (𝑣 = 𝑧 → ⟨𝑣, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
1918eqeq2d 2770 . . . . . . 7 (𝑣 = 𝑧 → (𝑤 = ⟨𝑣, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
20 sbequ 2513 . . . . . . . 8 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
21 cbvopab1.2 . . . . . . . . 9 𝑥𝜓
22 cbvopab1.3 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑𝜓))
2321, 22sbie 2545 . . . . . . . 8 ([𝑧 / 𝑥]𝜑𝜓)
2420, 23syl6bb 276 . . . . . . 7 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑𝜓))
2519, 24anbi12d 749 . . . . . 6 (𝑣 = 𝑧 → ((𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2625exbidv 1999 . . . . 5 (𝑣 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2716, 17, 26cbvex 2417 . . . 4 (∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2811, 27bitri 264 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2928abbii 2877 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
30 df-opab 4865 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31 df-opab 4865 . 2 {⟨𝑧, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
3229, 30, 313eqtr4i 2792 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wnf 1857  [wsb 2046  {cab 2746  cop 4327  {copab 4864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865
This theorem is referenced by:  cbvopab1v  4878  cbvmptf  4900  cbvmpt  4901  phpreu  33706  poimirlem26  33748  mbfposadd  33770
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