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Theorem cbvralf 3416
 Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2391. Use the weaker cbvralfw 3414 when possible. (Contributed by NM, 7-Mar-2004.) (Revised by Mario Carneiro, 9-Oct-2016.) (New usage is discouraged.)
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 1916 . . . 4 𝑧(𝑥𝐴𝜑)
2 cbvralf.1 . . . . . 6 𝑥𝐴
32nfcri 2968 . . . . 5 𝑥 𝑧𝐴
4 nfs1v 2161 . . . . 5 𝑥[𝑧 / 𝑥]𝜑
53, 4nfim 1898 . . . 4 𝑥(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
6 eleq1w 2894 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
7 sbequ12 2254 . . . . 5 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
86, 7imbi12d 348 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝜑) ↔ (𝑧𝐴 → [𝑧 / 𝑥]𝜑)))
91, 5, 8cbvalv1 2362 . . 3 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑))
10 cbvralf.2 . . . . . 6 𝑦𝐴
1110nfcri 2968 . . . . 5 𝑦 𝑧𝐴
12 cbvralf.3 . . . . . 6 𝑦𝜑
1312nfsb 2566 . . . . 5 𝑦[𝑧 / 𝑥]𝜑
1411, 13nfim 1898 . . . 4 𝑦(𝑧𝐴 → [𝑧 / 𝑥]𝜑)
15 nfv 1916 . . . 4 𝑧(𝑦𝐴𝜓)
16 eleq1w 2894 . . . . 5 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
17 sbequ 2091 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
18 cbvralf.4 . . . . . . 7 𝑥𝜓
19 cbvralf.5 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
2018, 19sbie 2545 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜓)
2117, 20syl6bb 290 . . . . 5 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
2216, 21imbi12d 348 . . . 4 (𝑧 = 𝑦 → ((𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2314, 15, 22cbvalv1 2362 . . 3 (∀𝑧(𝑧𝐴 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
249, 23bitri 278 . 2 (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜓))
25 df-ral 3131 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
26 df-ral 3131 . 2 (∀𝑦𝐴 𝜓 ↔ ∀𝑦(𝑦𝐴𝜓))
2724, 25, 263bitr4i 306 1 (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐴 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  Ⅎwnf 1785  [wsb 2070   ∈ wcel 2115  Ⅎwnfc 2958  ∀wral 3126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-10 2146  ax-11 2162  ax-12 2178  ax-13 2391 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clel 2892  df-nfc 2960  df-ral 3131 This theorem is referenced by:  cbvrexf  3417  cbvral  3422
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