Theorem rmoim 3670

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
rmoim	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))

Proof of Theorem rmoim

Step	Hyp	Ref	Expression
1		df-ral 3112	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		imdistan 568	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
3	2	albii 1805	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
4	1, 3	bitri 276	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
5		moim 2582	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6		df-rmo 3115	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-rmo 3115	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8	5, 6, 7	3imtr4g 297	. 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
9	4, 8	sylbi 218	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1523   ∈ wcel 2083  ∃*wmo 2576  ∀wral 3107  ∃*wrmo 3110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-mo 2578  df-ral 3112  df-rmo 3115

This theorem is referenced by:  rmoimia  3671  reuimrmo  3675  2rmorex  3684  2reurex  3689  disjss2  4939  catideu  16779  evlseu  19987  frlmup4  20631  2ndcdisj  21752  2sqreulem1  25708  2sqreunnlem1  25711  poimirlem18  34462  poimirlem21  34465