Theorem moim 2100

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 1551	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		ax-4 1520	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3		spsbim 1853	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4	2, 3	anim12d 335	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → (𝜓 ∧ [𝑦 / 𝑥]𝜓)))
5	4	imim1d 75	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
6	5	alimdv 1889	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
7	1, 6	alimd 1531	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8		ax-17 1536	. . 3 ⊢ (𝜓 → ∀𝑦𝜓)
9	8	mo3h 2089	. 2 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
10		ax-17 1536	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
11	10	mo3h 2089	. 2 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
12	7, 9, 11	3imtr4g 205	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1361  [wsb 1772  ∃*wmo 2037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040

This theorem is referenced by:  moimi  2101  euimmo  2103  moexexdc  2120  euexex  2121  rmoim  2950  rmoimi2  2952  ssrmof  3230  disjss1  3998  reusv1  4470  funmo  5243  uptx  14045