Theorem moim 2109

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 1555	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		ax-4 1524	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3		spsbim 1857	. . . . . 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 1893	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
7	1, 6	alimd 1535	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)))
8		ax-17 1540	. . 3 ⊢ (𝜓 → ∀𝑦𝜓)
9	8	mo3h 2098	. 2 ⊢ (∃*𝑥𝜓 ↔ ∀𝑥∀𝑦((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
10		ax-17 1540	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
11	10	mo3h 2098	. 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 1362  [wsb 1776  ∃*wmo 2046

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by:  moimi  2110  euimmo  2112  moexexdc  2129  euexex  2130  rmoim  2965  rmoimi2  2967  ssrmof  3246  disjss1  4016  reusv1  4493  funmo  5273  uptx  14510