Mathbox for Jonathan Ben-Naim < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1109 Structured version   Visualization version   GIF version

Theorem bnj1109 32286
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1109.1 𝑥((𝐴𝐵𝜑) → 𝜓)
bnj1109.2 ((𝐴 = 𝐵𝜑) → 𝜓)
Assertion
Ref Expression
bnj1109 𝑥(𝜑𝜓)

Proof of Theorem bnj1109
StepHypRef Expression
1 bnj1109.2 . . . . . . 7 ((𝐴 = 𝐵𝜑) → 𝜓)
21ex 416 . . . . . 6 (𝐴 = 𝐵 → (𝜑𝜓))
32a1i 11 . . . . 5 ((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
43ax-gen 1797 . . . 4 𝑥((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
5 bnj1109.1 . . . . 5 𝑥((𝐴𝐵𝜑) → 𝜓)
6 impexp 454 . . . . . 6 (((𝐴𝐵𝜑) → 𝜓) ↔ (𝐴𝐵 → (𝜑𝜓)))
76exbii 1849 . . . . 5 (∃𝑥((𝐴𝐵𝜑) → 𝜓) ↔ ∃𝑥(𝐴𝐵 → (𝜑𝜓)))
85, 7mpbi 233 . . . 4 𝑥(𝐴𝐵 → (𝜑𝜓))
9 exintr 1893 . . . 4 (∀𝑥((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓))) → (∃𝑥(𝐴𝐵 → (𝜑𝜓)) → ∃𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓)))))
104, 8, 9mp2 9 . . 3 𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓)))
11 exancom 1862 . . 3 (∃𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓))) ↔ ∃𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))))
1210, 11mpbi 233 . 2 𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓)))
13 df-ne 2952 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
1413imbi1i 353 . . 3 ((𝐴𝐵 → (𝜑𝜓)) ↔ (¬ 𝐴 = 𝐵 → (𝜑𝜓)))
15 pm2.61 195 . . . 4 ((𝐴 = 𝐵 → (𝜑𝜓)) → ((¬ 𝐴 = 𝐵 → (𝜑𝜓)) → (𝜑𝜓)))
1615imp 410 . . 3 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (¬ 𝐴 = 𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1714, 16sylan2b 596 . 2 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1812, 17bnj101 32221 1 𝑥(𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ≠ wne 2951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-ne 2952 This theorem is referenced by:  bnj1030  32487  bnj1128  32490  bnj1145  32493
 Copyright terms: Public domain W3C validator