ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvralf GIF version

Theorem cbvralf 2689
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.)
Hypotheses
Ref Expression
cbvralf.1 𝑥𝐴
cbvralf.2 𝑦𝐴
cbvralf.3 𝑦𝜑
cbvralf.4 𝑥𝜓
cbvralf.5 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvralf (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)

Proof of Theorem cbvralf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1521 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralf.1 . . . . . 6 𝑥𝐴
32nfcri 2306 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 1932 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfim 1565 . . . 4 𝑥(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
6 eleq1 2233 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 1764 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7imbi12d 233 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜑)))
91, 5, 8cbval 1747 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑))
10 cbvralf.2 . . . . . 6 𝑦𝐴
1110nfcri 2306 . . . . 5 𝑦 𝑧𝐴
12 cbvralf.3 . . . . . 6 𝑦𝜑
1312nfsb 1939 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfim 1565 . . . 4 𝑦(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
15 nfv 1521 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1 2233 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 1833 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralf.4 . . . . . . 7 𝑥𝜓
19 cbvralf.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 1784 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20bitrdi 195 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21imbi12d 233 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbval 1747 . . 3 (∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
249, 23bitri 183 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
25 df-ral 2453 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
26 df-ral 2453 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 211 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1346  wnf 1453  [wsb 1755  wcel 2141  wnfc 2299  wral 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453
This theorem is referenced by:  cbvral  2692  ffnfvf  5655
  Copyright terms: Public domain W3C validator