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Theorem bnj1109 31174
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 399 . . . . . 6 (𝐴 = 𝐵 → (𝜑𝜓))
32a1i 11 . . . . 5 ((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
43ax-gen 1877 . . . 4 𝑥((𝐴𝐵 → (𝜑𝜓)) → (𝐴 = 𝐵 → (𝜑𝜓)))
5 bnj1109.1 . . . . 5 𝑥((𝐴𝐵𝜑) → 𝜓)
6 impexp 439 . . . . . 6 (((𝐴𝐵𝜑) → 𝜓) ↔ (𝐴𝐵 → (𝜑𝜓)))
76exbii 1933 . . . . 5 (∃𝑥((𝐴𝐵𝜑) → 𝜓) ↔ ∃𝑥(𝐴𝐵 → (𝜑𝜓)))
85, 7mpbi 221 . . . 4 𝑥(𝐴𝐵 → (𝜑𝜓))
9 exintr 1981 . . . 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 1947 . . 3 (∃𝑥((𝐴𝐵 → (𝜑𝜓)) ∧ (𝐴 = 𝐵 → (𝜑𝜓))) ↔ ∃𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))))
1210, 11mpbi 221 . 2 𝑥((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓)))
13 df-ne 2975 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
1413imbi1i 340 . . 3 ((𝐴𝐵 → (𝜑𝜓)) ↔ (¬ 𝐴 = 𝐵 → (𝜑𝜓)))
15 pm2.61 183 . . . 4 ((𝐴 = 𝐵 → (𝜑𝜓)) → ((¬ 𝐴 = 𝐵 → (𝜑𝜓)) → (𝜑𝜓)))
1615imp 395 . . 3 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (¬ 𝐴 = 𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1714, 16sylan2b 583 . 2 (((𝐴 = 𝐵 → (𝜑𝜓)) ∧ (𝐴𝐵 → (𝜑𝜓))) → (𝜑𝜓))
1812, 17bnj101 31108 1 𝑥(𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1635   = wceq 1637  wex 1859  wne 2974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894
This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1860  df-ne 2975
This theorem is referenced by:  bnj1030  31372  bnj1128  31375  bnj1145  31378
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