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Theorem cbvopab1 4646
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 1829 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
2 nfv 1829 . . . . . . 7 𝑥 𝑤 = ⟨𝑣, 𝑦
3 nfs1v 2421 . . . . . . 7 𝑥[𝑣 / 𝑥]𝜑
42, 3nfan 1815 . . . . . 6 𝑥(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
54nfex 2138 . . . . 5 𝑥𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
6 opeq1 4331 . . . . . . . 8 (𝑥 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑣, 𝑦⟩)
76eqeq2d 2616 . . . . . . 7 (𝑥 = 𝑣 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑣, 𝑦⟩))
8 sbequ12 2095 . . . . . . 7 (𝑥 = 𝑣 → (𝜑 ↔ [𝑣 / 𝑥]𝜑))
97, 8anbi12d 742 . . . . . 6 (𝑥 = 𝑣 → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
109exbidv 1836 . . . . 5 (𝑥 = 𝑣 → (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)))
111, 5, 10cbvex 2255 . . . 4 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑))
12 nfv 1829 . . . . . . 7 𝑧 𝑤 = ⟨𝑣, 𝑦
13 cbvopab1.1 . . . . . . . 8 𝑧𝜑
1413nfsb 2424 . . . . . . 7 𝑧[𝑣 / 𝑥]𝜑
1512, 14nfan 1815 . . . . . 6 𝑧(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
1615nfex 2138 . . . . 5 𝑧𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑)
17 nfv 1829 . . . . 5 𝑣𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)
18 opeq1 4331 . . . . . . . 8 (𝑣 = 𝑧 → ⟨𝑣, 𝑦⟩ = ⟨𝑧, 𝑦⟩)
1918eqeq2d 2616 . . . . . . 7 (𝑣 = 𝑧 → (𝑤 = ⟨𝑣, 𝑦⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
20 sbequ 2360 . . . . . . . 8 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑))
21 cbvopab1.2 . . . . . . . . 9 𝑥𝜓
22 cbvopab1.3 . . . . . . . . 9 (𝑥 = 𝑧 → (𝜑𝜓))
2321, 22sbie 2392 . . . . . . . 8 ([𝑧 / 𝑥]𝜑𝜓)
2420, 23syl6bb 274 . . . . . . 7 (𝑣 = 𝑧 → ([𝑣 / 𝑥]𝜑𝜓))
2519, 24anbi12d 742 . . . . . 6 (𝑣 = 𝑧 → ((𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2625exbidv 1836 . . . . 5 (𝑣 = 𝑧 → (∃𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)))
2716, 17, 26cbvex 2255 . . . 4 (∃𝑣𝑦(𝑤 = ⟨𝑣, 𝑦⟩ ∧ [𝑣 / 𝑥]𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2811, 27bitri 262 . . 3 (∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓))
2928abbii 2722 . 2 {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
30 df-opab 4635 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
31 df-opab 4635 . 2 {⟨𝑧, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜓)}
3229, 30, 313eqtr4i 2638 1 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wnf 1698  [wsb 1866  {cab 2592  cop 4127  {copab 4633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-opab 4635
This theorem is referenced by:  cbvopab1v  4649  cbvmptf  4667  cbvmpt  4668  phpreu  32363  poimirlem26  32405  mbfposadd  32427
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