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Theorem ralcom2 2976
Description: Commutation of restricted universal quantifiers. Note that 𝑥 and 𝑦 need not be distinct (this makes the proof longer). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
ralcom2 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem ralcom2
StepHypRef Expression
1 eleq1 2571 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21sps 1996 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
32imbi1d 326 . . . . . . . . 9 (∀𝑥 𝑥 = 𝑦 → ((𝑥𝐴𝜑) ↔ (𝑦𝐴𝜑)))
43dral1 2208 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥𝐴𝜑) ↔ ∀𝑦(𝑦𝐴𝜑)))
54bicomd 208 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∀𝑦(𝑦𝐴𝜑) ↔ ∀𝑥(𝑥𝐴𝜑)))
6 df-ral 2796 . . . . . . 7 (∀𝑦𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴𝜑))
7 df-ral 2796 . . . . . . 7 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
85, 6, 73bitr4g 298 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
92, 8imbi12d 329 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝑥𝐴 → ∀𝑦𝐴 𝜑) ↔ (𝑦𝐴 → ∀𝑥𝐴 𝜑)))
109dral1 2208 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥(𝑥𝐴 → ∀𝑦𝐴 𝜑) ↔ ∀𝑦(𝑦𝐴 → ∀𝑥𝐴 𝜑)))
11 df-ral 2796 . . . 4 (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐴 𝜑))
12 df-ral 2796 . . . 4 (∀𝑦𝐴𝑥𝐴 𝜑 ↔ ∀𝑦(𝑦𝐴 → ∀𝑥𝐴 𝜑))
1310, 11, 123bitr4g 298 . . 3 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝐴𝑦𝐴 𝜑 ↔ ∀𝑦𝐴𝑥𝐴 𝜑))
1413biimpd 214 . 2 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑))
15 nfnae 2201 . . . . 5 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
16 nfra2 2825 . . . . 5 𝑦𝑥𝐴𝑦𝐴 𝜑
1715, 16nfan 2064 . . . 4 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑)
18 nfnae 2201 . . . . . . . 8 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
19 nfra1 2820 . . . . . . . 8 𝑥𝑥𝐴𝑦𝐴 𝜑
2018, 19nfan 2064 . . . . . . 7 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑)
21 nfcvf 2669 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
2221adantr 474 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) → 𝑥𝑦)
23 nfcvd 2647 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) → 𝑥𝐴)
2422, 23nfeld 2654 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) → Ⅎ𝑥 𝑦𝐴)
2520, 24nfan1 2063 . . . . . 6 𝑥((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) ∧ 𝑦𝐴)
26 rsp2 2815 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐴 𝜑 → ((𝑥𝐴𝑦𝐴) → 𝜑))
2726ancomsd 463 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴 𝜑 → ((𝑦𝐴𝑥𝐴) → 𝜑))
2827expdimp 446 . . . . . . 7 ((∀𝑥𝐴𝑦𝐴 𝜑𝑦𝐴) → (𝑥𝐴𝜑))
2928adantll 737 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) ∧ 𝑦𝐴) → (𝑥𝐴𝜑))
3025, 29ralrimi 2835 . . . . 5 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) ∧ 𝑦𝐴) → ∀𝑥𝐴 𝜑)
3130ex 443 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) → (𝑦𝐴 → ∀𝑥𝐴 𝜑))
3217, 31ralrimi 2835 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ∀𝑥𝐴𝑦𝐴 𝜑) → ∀𝑦𝐴𝑥𝐴 𝜑)
3332ex 443 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑))
3414, 33pm2.61i 171 1 (∀𝑥𝐴𝑦𝐴 𝜑 → ∀𝑦𝐴𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wa 378  wal 1466  wcel 1937  wnfc 2633  wral 2791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485
This theorem depends on definitions:  df-bi 192  df-an 380  df-tru 1471  df-ex 1693  df-nf 1697  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ral 2796
This theorem is referenced by:  tz7.48lem  7235  imo72b2  36975  tratrb  37253  tratrbVD  37606
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