Theorem moim 1964

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moim	⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem moim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1434	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		ax-4 1400	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3		spsbim 1724	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4	2, 3	anim12d 318	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
5	4	imim1d 69	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
6	5	alimdv 1759	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
7	1, 6	alimd 1414	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8		ax-17 1419	. . 3 ⊢ (𝜓 → ∀𝑦𝜓)
9	8	mo3h 1953	. 2 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
10		ax-17 1419	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
11	10	mo3h 1953	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
12	7, 9, 11	3imtr4g 194	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241  [wsb 1645  ∃*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  moimi  1965  euimmo  1967  moexexdc  1984  euexex  1985  rmoim  2740  rmoimi2  2742  disjss1  3751  reusv1  4190  funmo  4917