MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sb6rfOLD Structured version   Visualization version   GIF version

Theorem 2sb6rfOLD 2490
Description: Obsolete version of 2sb6rf 2489 as of 13-Apr-2023. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2sb5rf.1 𝑧𝜑
2sb5rf.2 𝑤𝜑
Assertion
Ref Expression
2sb6rfOLD (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Distinct variable group:   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb6rfOLD
StepHypRef Expression
1 sbequ12r 2244 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2 sbequ12r 2244 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
31, 2sylan9bb 510 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
43pm5.74i 272 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
542albii 1812 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
6 2sb5rf.2 . . . . 5 𝑤𝜑
7619.23 2201 . . . 4 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
87albii 1811 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
9 2sb5rf.1 . . . 4 𝑧𝜑
10919.23 2201 . . 3 (∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
118, 10bitri 276 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
12 2ax6e 2486 . . 3 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
13 pm5.5 363 . . 3 (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑))
1412, 13ax-mp 5 . 2 ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑)
155, 11, 143bitrri 299 1 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wex 1771  wnf 1775  [wsb 2060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator