Theorem rmoim 3688

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 3055	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)))
2		imdistan 572	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)))
3	2	albii 1826	. . 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 2548	. . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)))
6		df-rmo 3345	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-rmo 3345	. . 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 1545   ∈ wcel 2119  ∃*wmo 2541  ∀wral 3054  ∃*wrmo 3344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-ral 3055  df-rmo 3345

This theorem is referenced by:  rmoimia  3689  reuimrmo  3693  2rmorex  3702  2reurex  3708  disjss2  5049  catideu  17639  rinvmod  19779  frlmup4  21783  evlseu  22066  2ndcdisj  23446  2sqreulem1  27434  2sqreunnlem1  27437  poimirlem18  38012  poimirlem21  38015