MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2sb6rf Structured version   Visualization version   GIF version

Theorem 2sb6rf 2553
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
Hypotheses
Ref Expression
2sb5rf.1 𝑧𝜑
2sb5rf.2 𝑤𝜑
Assertion
Ref Expression
2sb6rf (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 2sb6rf
StepHypRef Expression
1 sbequ12r 2223 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2 sbequ12r 2223 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
31, 2sylan9bb 738 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
43pm5.74i 260 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
542albii 1861 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
6 2sb5rf.2 . . . . 5 𝑤𝜑
7619.23 2191 . . . 4 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
87albii 1860 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
9 2sb5rf.1 . . . 4 𝑧𝜑
10919.23 2191 . . 3 (∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
118, 10bitri 264 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
12 2ax6e 2551 . . 3 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
13 pm5.5 350 . . 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 287 1 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1594  wex 1817  wnf 1821  [wsb 2010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator